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FIRST STEPS 


AMONG. _ Bréures. 


A ae ag 7 Fundamental Rules 
of Arithmetic. 


TEACHERS ION. 


We / 


- sts B 
BE UPN . BEEBE, | He 


WV, 
/ CANANDAIGUA, N. Y.. 


SEVENTH EDITION, ENLARGED AND CAREFULLY REVISED, 


SYRACUSE, N. Y.: 
C. W. BARDEEN, PUBLISHER. 
1884. Sas 
, ny 


Lee ey 


oo 
" 


ost : 


ee Libran 
ee 


PREFACE TO TEACHERDS’? 
EDITION. 


In putting this work before the-public the 
author disclaims any ambitious schemes or 
“reat expectations,” but he wishes to have 
the book for the use of his assistant teachers 
both as to methods and examples. The author 
has used some parts of it for many years and 
feels confident that excellent results may be 
obtained by using it. 

The aim of the book is to give so much 
practice as to fix each method in the pupil’s 
inind, rather than to deal with the philosophy 
of each operation. leaving any teacher who 
believes that no step should be taken unless 
the pupil understands the reasoning process 
by which that step may be reached, to give it 
in his own way. It is possible that a few who 
see this book may have found that 7 times 8 
are 56 by actual addition, yet those who have 
never added it may know the fact just as well 
for all practical purposes. 

If no one were to eat until he understood 
how food nourishes the system there would be 
a deal of hunger in the world. 


4 PREFACE TO TEACHERS’ EDITION. 


This book deals only with the fundamental 
rules of arithmetic. ‘The intention is that they 
shall be so thoroughly mastered that much less 
time will be required for the remainder of the 
subject of arithmetic than would otherwise 
be needed. 

The teacher is to use the Teachers’ Edition © 
for one to two years before the pupil has 
advanced enough to use the Pupils’ Edition or 
in fact any book on arithmetic. It is recom- 
mended that teachers begin to teach numbers 
as given in the first part of this book after 
pupils who have the alphabet and words to 
learn have been in school four to six months. 

In each new operation the examples are 
very easy ; as more problems are given they 
gradually increase in difficulty. By teaching 
the four operations of addition, subtraction, 
multiplication, and division, from the first, 
the examples are of such a kind as to compel 
some thoughtfulness on the part of the pupil. 

Much pains has been taken to make exam- 
ples of a sort to interest the youngest pupils. 

Those teachers who wish to. teach only 
addition and subtraction at first can designate 
those examples involving multiplication or 
division by some mark, and omitting them ~ 
may return to them afterwards and so secure 
the variety of examples so essential to a pupil’s 
real progress. It has been found, however, 
by actual trial. that pupils may learn the four 
operations from the first without serious 
difficulty. 


PREFACE TO TEACHERS’ EDITION. é 


— 


The first pages are devoted to what is known 
as the “ Grube Method.” If the teacher pre- 
fers it, the schedules may be omitted, and, in 
passing through the first time, the multiplica- 
tion and division also, as before stated. 

The author hopes that teachers into whose 
hands this work may come will give it a 
thorough examination. Special attention is 
called to the treatment of numeration and 
notation. The examples are not all given in 
one place, to be forgotten, but are so placed as 
to review the subject often. 

Attention is called to the examples for rapid 
solving and the illustration of the easy exam- 
ples given under each rule. Also to the 
method of teaching long division and to the 
definition of addition. . 

The method of teaching the addition, sub- 
traction, multiplication and division tables is 
believed to be entirely new, so far as being 
published is concerned. The author dis- 
covered and used the method about ten years 
ago, and in his school has found it invaluable. 

To hear a recitation of a large class in 
tables and make the questions to each pupil 
promiscuous, and yet fuil enough to satisfy the 
teacher that the pupil has a thorough knowl- 
edge of the tables gone over, is not only very 
wearying to the teacher but is exceedingly 
dificult also. By the old method a pupil 
frequently acquires the habit of saying the 


6 PREFACE TO TEACHERS’ EDITION. 


table from the beginning to find the result of 
any combination, as 7 times 6. 

To enable any one to make new series like 
those here given, I insert the method. The 
following is for g’s and review. In the given 
lines of ‘figures there is one more figure in the 
upper line than in the lower one: 


3.94 56 ORS 
4 5 On aRO mG 
If the upper line be written several times 
and the lower line in the same way as follows: 


3.4 5.6.7 ° 8 9 3345 Oyo 
4 5 6-7-8 9. 4°95 07. 3 ee 
3.4 °5.6 9°83 9 - 3°45) 0 ee 
6° 7 8 9 4 5 6 .Y 626 4 oe 
3.4 5 6 7.8 9 3 4.95 G6 eRe 
8.9 4.5 6 7 8) 9 4° 5y 0 ye 


The first 9 in the lower line comes one 
place before the 9 in -the: upper lines sia. 
second g in the lower line comes two places 
before the second g in the upper line, and so 
on until it has been under every figure i in the 
upper line. If. written farther, the series will 
be repeated as shown above, where 4 and 3 
occur again at the end as they did at the 
beginning. The upper line of figures must be 
written one less number of times than there 
are figures In it. 

This series may be used for addition or 
multiplication, thus: 4 and 3 are 7, 5 and 4 are 
9, &c., or 4 times 3 are 12, 5 times 4 are 20, &c, 


PREFACE TO TEACHERS’ EDITION. . 7 


This arrangement is objectionable for most 
of the results in addition vary only by 2 or 4 
and are not as promiscuous as they should be. 
By disarranging the upper line of figures we 
have 6, 9, 5, 8, 4, 7, 3. Re-writing this for the 
upper line and writing the lower line as before 
we have 


momo. 47° 3° 6 9-5 8 4:7 3 
peer 7) oO 4. 5 6 78 Oo 4 5 
nse) 42707360 9° 5 8 47 3 
Meee 4 5-6-7 8. O04 5 67 
Ppeseeo 4.07.3) 6955.8 4.7 3 
Bepcmeaees O72 8 9 4°75 6.7 8-9 


which, like the other series, contains every 
combination between 4 and 3 and g and 9g 
inclusive, and none repeated except by inver- 
sion as 4+5 and 5+4, but unlike that series 
it is entirely promiscuous. 

To make a series for subtraction, write the 
series as above, and write the sums of the 
numbers above, thus: 

Pominertets P2207 11.15 12 16 13° &c. 
Sogo 401 °3° 6 9 5.8 4°73 &e. 
ieperc 7) 5 9° 4-5 6 7 8 9. &e. 

Then copy for the minuends the upper num- 
bers, and for the subtrahends the lower ones 
and the series becomes 


Soi aei i i5-t2 16° 7 11 15-12 16.13°° &c. 

mesmo eS. 94.5560 °7 & 9 &c. 

In this book those for subtraction have been 
still further disarranged so that the results 


8 | PREFACE TO TEACHERS’ EDITION. 


willnot be 695 84 7 3 and soon lest the pupils 
notice it and recite that instead of subtracting. 
For division find the products for the upper 
line instead of the sums. 

For division with remainders, which is an 
excellent preparation for short division, after 
having written the products above as before, 
add to each one of them a number less than 
the lower number in that column and write 
for the upper line these sums and for the 
lower line the lowest line of figures. 

In the first series of division with remain- 
ders, the remainders are very small, that it 
may be as easy as possible, 

In the first series the combinations do not 
go as far as g— that is 2 and g, 9 times 2, &c., 
but only to combinations of 2 3 4 5 and 6 with 
2 3 4 5 and 6. The examples which follow 
immediately after the learning of any table 
involve only what is contained in the table. 

It is believed there is a very large amount of 
work for practice, both in the Pupils’ Edition 
and in the Teachers’ Edition, more than twice 
as much as in other works of the kind. The 
greatest care has been taken that they may 
proceed from the easiest to those involving 
every difficulty which pupils should meet at 
the age for which this book is designed. 

As anything is learned it is immediately 
put into use. 

; LEVI N. BEEBE 


Canandaigua, N. Y., July, 1877. 


_ FIRST STEPS AMONG FIGURES. 9 


Aur oa. gd Bee 
ie eet Ls 1 


ONE, 
(See Appendix, page 185.) 

1. Be sure the pupil has the idea of ove thing 
in distinction from ¢wo or more things. 

Illustrate by objects as much as possible, 
using small sticks, or square blocks % inch 
square and ¢ inch thick of different colors, or 
bright cents. Only ten of each are needed and 
if the teacher has a// these he can add interest 
to the exercises. 

An abacus, or numeral frame, is almost indis- 
pensable as a further help. 

Show the pupil that taking one article (as a 
bean, a cent or a block) one time, or putting it 
into a box or upon a book or table makes one 
article there, which is the interpretation of 
“once one is one.” —Let the fupz/ place the 
article, and thus impress his mind more thor- 
oughly with the idea once 1 is 1, written-1 xX 1 
=Tty 

*In this book the multiplier is uniformly placed on the 
right of the sign XX: thus 2 times one are 2 will be written 
[oR es ie 


See Appendix, pp. 185-192, 


id FIRST STEPS AMONG FIGURES. 


2. The idea of division may be taught in the 
following way: The teacher may place a pile 
of 2 blocks on a table or book and ask the 
pupil “ How many times one block have I in 
this pile?” Pupil: ‘“‘Two times.” Teacher: 
‘One block in two blocks how many times?” 
Pupil: “Two times.” Teacher: ‘One in two 
how many times?” Pupil: “Two times.” 
(The pupil may use in each answer the word 
“twice” instead of the words “two times.”) 
Teach that this is written 2+-1=2 and should 
be read by the youngest pupils, 1 in 2 twice. 


* Schedule : 
o | tX1=1. (Read once one is one.) 
| I-+1=1. (Read one in one, once.) 
What can you find once in the school-room, 
in your pocket, on your face, at home? &c. 
What is there that moves on one wheel? 


TWO. 
3. Teach in counting that the second of two 
things is not of itself tzevo, but one. 
In teaching number and in operations on 
numbers use objects for some time —at least 


*The schedules being written on the blackboard, the pupils 
are to be taught to read them, and eventually to make them 
themselves, 


FIRST’STEPS AMONG FIGURES. II 


three months to six months, until the pupil is 
thoroughly familiar with the composition of 
numbers. 

4. Teach pupils to count to 11 and continue 
to teach counting daily until the pupil can count 
100. 

Schedule : 
o | 1+1=2. (Read one and one are two.) 
0 1x2=2. (Read twice one are two.) 
| 2—1=1. (Read one from two leaves one.) 
| 2-1=2. (Read one in two twice.) 

2 is one more than what number? 

I is one less than what number? 

2 is the double of what number? 

2 is twice what number? 

1 is one-half of what number ? 

1 and 1 are*? 1 from 2 leaves? 

(1 from 2 leaves 1, because 1 and 1 are 2.) 

Mary has 2 sticks of candy ; she gives away 
2 sticks ; how many sticks has she left ? 2 from 
2 leaves? 

Henry had 2 marbles ; he gives none away: 
how many has he? Nothing from 2 leaves ? 


or ‘‘how much” in such examples, according to the 


sense. 
Show the pupils that 1 block placed on the table, and 


then another, make 2 blocks there, hence 2 is 2 times I. 


12 FIRST STEPS AMONG FIGURES. 


What is there that moves on two wheels? 

Hold up two fingers. 

What have you on your head of which there 
are 2 and only 2? In the school room? At 
home? &c. 

What animals walk on 2 legs? 

5. What is 2 of an apple? (Let the pupil 
take an apple and cut it into halves and ask 
him what one piece is called. Show him that 
if he takes one-half of the apple there is left 
as much as he takes.) What is 3 of 2 apples? 
Placing 2 apples on the table, let one of the 
pupils take half of them by leaving as much as 
he takes. 

(Vary the exercise by taking 2 of a stick of 
candy, 3 of 2 sticks, &c ) 

6. 2 is the double of what number? Of what 
number is 1 one-half? 

What number must I double to get 2? 

I know a number that is 1 more than 1 ; 
what number is it? 

7. What number must be added to 1 to get 
2? Fred had 2 dimes and bought peaches with 
1 dime. How many dimes had he left? 

(No analysis of these examples is expected ; 
simply a prompt answer.) 


FIRST STEPS AMONG FIGURES. 1 


A slate pencil costs t cent, how much will 2 
slate pencils cost? 

Charles* had a marble, and his sister had 
twice as many. How many did she have? 

How many slate pencils can you buy for 2 
cents? 

How many 2-cent stamps can you buy for 2 
cents? How many ti-cent stamps? 

(Both these and the following examples 
should be gone over many times, taking them 
in a different order each time and often giving 
them promiscuously. ) 

8. Teach that there are 2 pints in a quart by 
pouring a pint cup full of water twice into a 
quart cup. ! 

What cost a quart of milk at 1 cent a pint? 


THREE. 
Schedule : 
9g. Measuring by 1. 
| Iet, ft 93) 
71 1+14+1=3. 
Pro xe = 3! ite 
71 3—1—1=1, for 3—1=2 and 2—1 
ie. 


*In such examples it will interest the class to use their 
names instead of those given, 


14 FIRST STEPS AMONG FIGURES, 


Measuring by 2. 
Tei p22 (2213: 
II 2X1+1=3. (To be réeadpencesg 
III 3 and 1 are 3, or once 2 plus I 
are 3.) 
3—2=1, 3—1=2. (To bereada2 
from 3 leaves 1 and 1 from 3 
leaves 2.) 
3--2=1 (and 1 rem.) (To be 
read 2 in 3 once and 1 re- 
mainder. ) 


The pupils should read these schedules many 
times each, until they are familiar with the lan- 
guage. 

1o. Illustrate by a pile of 3 blocks. How 
many times have I 2 blocks in the pile? Once. 
Take them away once then. How many are 
left, or how many remain? One. 2 blocks in 
3 blocks how many times? Once and 1 re- 
mainder. 2 in 3 how many times? Once and 
I remainder. 

11. To be written on the blackboard for 
pupils to bring written with the answers to reci- 
tation. 

1 Loe 3—1+1=? 

I+1=? I+2=? 24+2=? 


FIRST STEPS AMONG FIGURES. 18 


g—2=—° 3-1 2—2=? 
2+1=? atit+i=? «m1X2+1=? 
2--2=? 3+2=? 373=? 


12. 3 is rt more than? 
1 is t less than? 
3 is 2 more than? 
2 is 1 less than? 
2 is t more than? 
1 is 2 less than? . 
3 is 3 times? 

13. To illustrate tell a pupil to put one block 
or one cent on the desk and then another. 
Show the pupils that a block has been put 
upon the desk twice and that there are two 
blocks there; hence 2 times 1 block are 2 
blocks ; also 2 times 1 orange are 2 oranges, 
and 2 times 1 pencil are 2 pencils, &c. 2 times 
any one thing are two of those things. 2 times 
I are 2. 

Show the pupils that 1 block taken 3 times 
or placed on a table 3 times makes 3 blocks 
there, hence 3 is 3 times 1. 

14. This form of illustration may be used fot 
any multiplication. 

How many pints in a quart ? 

Teach pupils to write numbers as high as 20, 


16 FIRST STEPS AMONG FIGURES. ~ 


It may be well to teach the writing of 12 before 
10 or 11. Show the pupil by the abacus or 
otherwise 12 objects and show him that they 
are 1 ten and 2 ones. Show him that we can- 
not write 12 by any ome of our figures ; then 
teach him about ten’s place and one’s place. 

Do not use the word units for several weeks 
yet. 2 
15. Teach pupils to count by 2’s from 2 to 6 
and back to 2,-thus: 2, 4, 6: ‘6; 4ie2. 

Explain that 1 and 1 are equal numbers, that 
is equal to each other ; 1 and 2 as well as 2 and 
3 are unequal numbers. 

16. Give the pupils much practice in exam- 
ples like the following: 3—1—1+12% Tobe 
read, how many are 3 less 1 less 1 plus 1? or 
3, subtract 1, subtract 1, add 1; or 3 minus 1 
minus 1 plus 1.* 


*These are to be read by the teacher, thus: 3, add 2, 
subtract 1, divide by 2, multiply by 3. 

The examples may be read through and those who can 
answer raise the hand; the teacher call upon one most — 
unlikely to be correct for the answer ; if incorrect call upon — 
another until the correct answer be given, 

It may be best at first and perhaps often to have the result 
of the first step given by one pupil, the next step by the next, 
&c EE. g. teacher, 3,add 2, 1st pupil says ‘‘5,” Teacher, 


FIRST STEPS AMONG FIGURES. 17 


2—1+2—1x1=? I+2—1—1x2+1=? 
2+1—2xX3—1+1=? 14+1X1—1+2-1=13 

Read with as much rapidity as the class can 
follow silently and give the answer at the end, 
‘the rapidity being increased as the pupils have 
more practice. : 

17, From what number can you take one 
and have one left ? ; 

Count by 2’s from 2 to Io. 

What number is twice 1 ? } | 

18. I write a number once, and again, to get 


23 what number did I write twice ? 


How many cents must you have to buy a 
3-cent stamp ? 
_ Mary had to get a pound of tea. for $1 ; her 
mother gave her $3 ; how much money ought 
she to bring back? 

Henry learned 1 line in his primer, and his 


? 


“subtract 1,”” 2d pupil says **4,”” Teacher, ‘divide by 2.” 3d 
pupil ‘¢ 2,” Teacher, ‘‘ multiply by 3.” 4th pupil ‘* 6,” Call 
on an inattentive pupil at any step in these examples for the 
answer. Usually, the teacher reads the whole example and 
the pupils give only the final answer, 

The foregoing examples are not written so as to be cor- 
rect for solving from the written or printed form for in that 
case 3-2 x 2 would mean, take 2 x 2 from 3, but it is to 
be read: 3 subtract 2, multiply by 2. 


2 


18 FIRST STEPS AMONG FIGURES. 


sister learned 1 line more than he did; how 
many did she learn? 

If 1 slate pencil cost 1 cent what will 3 slate 
pencils cost ? 

Anna found 3 roses in the garden; how can 
she divide them between her father and mother ? 

Can she give them an equal number ? 

How many roses must she have had in order 
to give her father 1, and her mother 1 also ? 

Count by 2’s from from 2 to 12. 

The counting may be sometimes in concert, 
oftener rst pupil say 2; 2d pupil, 4; 3d, 6 and 
so on, and perhaps oftenest one pupil give the 
whole series. 

Teach pupils to count rst, 2d, 3d, &c. 


FOUR. 
Schedule : 


20. Measuring by 1. 
ES) as OM 


71 1+1+1+1=4. (Because 1+1=2, 
2+1=3, 3+1=4.) 

7s 1X 4=>—4. 

7 it 4—-1—1—1=1,0r 4—-I—I—1I—1=0. 

7 i 471=4. 

44 


FIRST STEPS AMONG FIGURES. 19 


Measuring by 2. 
fetee2. §2-+2—. 
2X 2=4. 
Ii 2. 4—-2=—2, or 4—2—2=0. 
Itt 40 4--2=2. 
Measuring by 3. 
III 3. 3+1=4,1+3=4. 
3Z34+1+1=5. 
ietie 4-3 —1,'4—I—. 


———_——— 


IIIr 4. 4+3=1 (and 1 remainder.) 


: a1 
3—2=? 2xX1+2=?2 4+3=? 
ai’? 3+1=? 3--2=? 
3—1I=? 2ti=? 3+1=? 
2+2=? Sem eet 4—3=? 
3-2=7? 2xX1+1=? 4-I=? 
4—2>=? 2x11? 4—4=? 
ae mad 4—2—2=? 
42>? 4+4=? 


22. Name animals with 4 legs ; with 2 legs. 
Name wagons and vehicles with 1 wheel ; 2 
wheels ; 3 wheels; 4 wheels. Compare them. 
(For instance a wagon with 4 wheels has how 
many more wheels than one with 2 wheels? &c.) 
23. 41S I more than ? 
t is 1 less than ? 


40) FIRST STEPS AMONG FIGURES, 


I more than ? 
1 less than ? 
1 more than ? 
2 less than ? 
r less than? 
2 more than ? 
2 less than? 
2 more than? 
1 is 1 more than? (Nothing.) 
4 is 4 times? 
24. Solve rapidly the following :. 
2X2—3+2xX1+1—2xX2=? 
4—I—1+1+1—3=how many less than 4? 
3—2+3—1—1xX2—1=how many times 1? 
I+2—1-+2+2—1=how many more than 2? 
Teach to count by 2’s from 2 to 20 and back 
to 2. 
25. I1X2—I X 3—2—2= how many less 
than 3? 
3—2+1X2—1—2+1=? 
4—2—1X3—1IX2—-1=? 
26. What number must I double to get 4? 
Of what number is 4 the double ? 
Of what number is 2 one-half? 
What number can be taken twice from 4? 
What number is 2 more than 1? 
What number must I add to 2 to get 4? 


KR b® NH HNO S&S FW DN 
w 


FIRST STEPS AMONG FIGURES. 21 


What number is 1-2 of four? 

How many less than 3 is the half of 4? 

27. Minnie had 4 pinks which she neglected 
sadly ; one day 1 of them withered, the second 
day another, and the following day 1 more. 
How many fresh ones had she then ? 

How many $’s are $2+ $2? 

How many apples are 3 apples and 2 apples ? 

28. Teach that there are 4 quarts in 1 gal- 
lon, taking a gallon measure and filling it by 
pouring a quart measure full of water into it 4 
times. 

Nellie bought a gallon of milk; how many 
quarts did she buy P 

She paid x dime for each quart; how many 
dimes did she pay for the gallon? 

If 2 qts. of milk cost 2 di, can you get a 
gal. for 3 di.? 

How much can you get for the 3 di.? 

If I drink a quart of milk in 2 days, what 
part of a qt. do I drink in 1 da.?— 

29. William having 4 apples, ate half of them 
and one more, how many had he left ? 

What number is 1 more than half of 4 ? 

Ann had 3 apples ; she gave an equal number 
to her mother, father and brother; how many 
did she give each ? 


22 FIRST STEPS AMONG FIGURES, 


Sarah cut 1 apple into 2 equal pieces; what 
would you call one of the pieces ? 

Teach to count by 2’s from 2 to 40 and back 
to 2. 

By marking off paper or pasteboard, or bet- 
ter a thin board, and cutting, according to the 
following directions, an excellent aid in teach- 
ing notation and numeration may be obtained. 
By ruling both ways, mark off into squares 10 
squares in a row and 21 rows, as shown below. 
Cut off one row or strip of 10 squares and then 
cut up the strip into single squares. After- 
wards cut off 10 strips of 10 squares, which 
will leave a large square containing 10 rows of 
small squares with 1o oe in each row. 


Babs 


| | 
Eek 
BEM eee 
BRE: 
(aaa 
| de ae 
Baba, 
Beha: 
Beeke 


The ro small squares cut up may be used to 
illustrate ones or units, and the strips, tens, 


FIRST STEPS AMONG FIGURES. 28 
while the large square will represent hundreds. 
With them it will be easy to show that 10 ones 
equal a ten, and ten tens one hundred, and the 
teacher will show where the ones, tens, and 
hundreds are written in numbers. 


FIVE. 
Schedule ; 


30. Measuring by r. 
ls 92S Be 


71 1+1+1+1+1=5. 
Aamee Tax 5S, 

71 5—1—I-1—-1=1, 
71 5+1=5. 

71 


Measuring by 2. 
Pele 2s 62 -b 2-95. 


II 2. 2x2+1=5. (See note p. 23.) 
I 2. 5—2—2=1. 
IIIT 5. 5-~-2=2 (1 remainder.) 
By 3 
IrIr 3. 3+2=5,2+3=5. 
It 2.- 3X1+2—5. - 
Prune 5— 42. 


5+3=1 (2.rem.) 


24 FIRST STEPS AMONG FIGURES. 


By 4. 

IIIs 4. 4+1=5, 1+4=56 

I Ir 4X1I+I1=5. 
IIIII = 5—4=1. 
5+4=1 (1 rem.) 
eT 

3—1=7 IX3+1=? 5—2—2=% 
5-2=? 2x2+1=7 5 ibaa 
4—2=12 5—2=% 5+5=—? 


Sy Gt gt+i1+i1=? ~ 5-4! 
A+3=? 4—2-—2=—1 3x1+2=3 
5—3=? 3+3=7? 4+1=? 
AA wood 5+4=7 2+2+1=? 
5+3=? 2+2=? 
32. 5 is I more than? 

2 is 1 less than? 

3 1s 2 less than? 

4 is 2 more than? 

What number added to 2 will make 5? 

5 is 4 more than ? 

3 is 1 more than? 

5 is 2 more than? 

2 is 2 less than? 

3 is 3 more than? 

1 is 2 less than ? 

5 is how many times 1? 


FIRST STEPS AMONG FIGURES. 25 


(Since 2 is added twice and 1 is added to the 
result to get 5, 2 times 2+1=5 and since 2 may 
be subtracted from 5 twice and 1 will remain, 
2 is contained in 5 twice and 1 remainder, or 
5-2=2 [1 rem. ].) 

33. Teach pupils to write and read Roman 
notation to V. Teach pupils to count by 2’s 
. from 2 to 50,and back to 2, and from 1 to 11 
and back. 

34. For rapid solving. 
5—2—3+2X2—-3+2=? Ans. 3. 
2X2+1—3X1X2—3+3=—?. Ans. 4. 
4—1+2—3X2—1—2X3—1=? Ans. 2. 
B-be--1—2X2—3--1=—!, Ans. o. 

5-3 +2—-3%3—1+2=?. Ans. 4, 
2+2—3xX2+3—2+1—2=? Ans. 2. 
2+1—2xX3—3+2X2 is how much more 
than 1 ? Ans. 3 more. 
3+2—1—2+1—2x3—1=? Ans.2 

Review these frequently. 

35. Review counting by 2’s. 

Teach to count by 2’s, commencing with 1, 
to 21. 

36. How many must I add to 3 to get 5? 
How many must be taken from 5 to get 3? 
Why? (Because z added to 3 makes 5.) 


26 FIRST STEPS AMONG FIGURES. 


37. How many times 2 must I add to 1 to get 
5? 

I have taken away twice 1 from a certain 
number and 2 remains. What number was it? 
_ I have taken 2 from a certain number and 1 
remains. What number was it? 

I have added 2 to a certain number and have 
3. What number was it? 

How many gallons are 2 quarts. Ans. None. 

John had 5 dimes ; he bought 2 copy books, 
each of which cost 2 dimes. How many dimes 
did he keep? (Illustrate, using dimes.) 

George read a lesson once. Helen read it as 
many times as he did and two times more. 
How many times did she read it? 

A father had 5 peaches and gave them to his 
3 children ; he gave the oldest 1 peach, and 
gave to each of the others an equal number ; 
how many did each of the younger children 
receive? 

A boy has 2 cents, he finds 2 cents; how 
many will he have to earn to have 5 cents? 

1 is 2 of what number? 

James has 5 marbles, he loses 2 ; how many 
more than 2 has he left ? 
2 boys are passing my house, and each boy 


FIRST STEPS AMONG FIGURES. 24 
is driving 2 goats ; how many goats are pass- 
ing my house? 

David rode 1 horse from the pasture to the 
barn and at the same time led 2 others ; how 
many horses did he bring to the barn? 

Jane had 5 chickens. A rat ate 1 of them, 
and then a cat ate half of what were left and 
I more ; how many lived? 

A boy, having 4 pockets, has 2 apples in 1 of 
them ; one pocket is empty, and he has r apple 
in each of the other pockets : how many apples 
has he? 

38. Teach counting by 2’s from 1 to 21 and 
back. Review former counting. (Do not 
teach all of this before giving other exercises, 
but require some of this kind of exercise daily 
until as much as is denoted above has been ac- 
complished. These directions apply to future 
countings. ) 


SIX, 
Schedule s 


39. Measuring by 1. (Teach pupils to make 
these schedules.) 
1 2 OE OS 


23° FIRST STEPS AMONG FIGURES. 


I «1+1+1+1+1+1=6. 
Tee tae O? 

I 6—1I—-1—iI—I—i=1, 
I 6+1=6. 

I 

I 


sn Box Uitnn, mg ee 


oy 
» SD 
Oo 


2 2+2+2=6. 

29 (2 iteew. 

2 6—2—2=2, 6—2—2—-2=0 
6 O24, 


Sra 2eb eee Ge 
UT 3 eee 0 eee 

6-0-3 350 3a 
Oz Gas 


Toad scl et As eet Greets 

Ii) 2.” xX +26, 
ZII3Iii 6. 6—4=2, 6—2==4: 

6+4=1 (2 rem.) 

By 5. 

Il: ir 5. S+1=6,1-+52e6- 
Tors 25 Xa reo. 

6. 6—5=1, 6—1=5, 6— 

5—I=o. 
6--5=1 (1 rem.) 


=a 


Se Carat Pes ke 


FIRST STEPS AMONG FIGURES. 29 


4O. 
B—2>?r 2X2+1=? 4-—2=>? 
ae eet kA fr e+t? 


axe 6 6-2? 6—2—2—2=? 
I+4+1? 3+2=? A-1—?. 
eee — es SIS oe xXe4+2=? 
aoe §.6-— 257 Roa t 
6—2—2=? 1xX5=—? 4—1—1—1==? 


41. 6 is 3 more than? 
What is = of 6? 
6 is 4+how many? 
6 is 2 times what number ? 
How many times can you take 4 from 6? 
6 is 3 times what number? 
4 less than 6 is? 
6isz+? 
What number is 3 less than 6? 
What number is half of 4? 
6isi1+? 
42. For rapid solving. 
It+2+2=? 3+2—1+2=—? 
Iti+i+2=—? 2+2+2=? 
3+2—-1+2x3=? 
§—24+3—2—3+1xX2+1—3+1? Ans. 3. 
eee tot Sl — 2X 2— 2 - aol 
Ans. 2. 3 


f, 


36 FIRST STEPS AMONG FIGURES. 


2+3—2xX2—5xX3+2—3xX1+2? Ans. 4. 
4—3+2-—-1X2+24+2+2+4+2+2+2? 


Ans. 16. 
5t+2+24+ 24+2424+24+2+2+2+42+42? 
Ans. 27. 
43. Count by 3’s from 3 to 12. Review 
counting. 


Count by 3’s from 3 to 18. Review the 
counting already taught. Teach Roman nota- 
tion to X. 

44. For rapid solving. 
6—2—2+3—2x2+2+2 7%? Ans. lo. 
2X3—4+3—1+2+2+2+2+2+2+2! 

Ans. 18. 
§—3t+1+3—2424+3+2+27242F2 
+2+2? Ans. 19. 
+3 —24+3+173+2+2+2+2+2=?% 

ANSo 12, 
5—2—2X2—14+4—3X34+3+34+2+2 

+2=% Ans. 18. 
2+3—24+37242 $2724 

2m ¢. Anse i197. 
4—3+2+2—3x2+2+2+/? Ans. 8. 
6+3+3—2 X 24343 +3 eee 

+22 Ans. 23. 

45. Examples in addition may be written on 


FIRST STEPS AMONG FIGURES. 31 


the board, a few for the pupils to solve each 
day on their slates and bring to class; and ad- 
ditional examples may be read to the class and 
solved on the slates or on the board.* 

46. 3 cts. is 2 cts. more than John has; how 
many has he? 

3 cts. is 1-2 of what money Jane has; how 
much has she P 

2 cts. is 3 cts. less than an orange cost; how 
many cts. did it cost? 

1 is how many less than 6? 


6 cts. will buy how many 3 cent. stamps? 2 
ct.stamps? 2ct. stamps? 5 ct. stamps? 


George had 1-2 of 6 cents; how many had 
he? 

Clara had 6 flowers; she gave them to her 
father and mother. If she gave each of them 
the same number, how many did her father 
get? Her mother? 

Charles had 6 cts.; he lost 2 of them; how 
many had he left P 

Carrie had 6 peaches: she gave her father 2 
of them and her mother 2; how many did she 
keep ? 

3 cts. is 3 cts. more than Byron’s money; 
how much money has he? 


" See T. Ed., p. 84, and P. Ed., p. 33. 


32 FIRST STEPS AMONG FIGURES, 


47. 3+2—1+2—5+3—2? Ans. 4. 

1-2 of 6+2—3 is what part of 4? Ans. 1-2. 

1+2+2—3—1+3—2=% Ans. 2. 

3+2—1—34+1X34343+3+2+2=? 
Ans. 19. 

6—4+2 —1X2-— 24243737 o eee 
2? Ans, 18. 

I put down a number once and again and 
again to get 6. What is the number ? 

48. Count by 3’s from 3 to 24. Review 
the counting. 

From what number can you take 2X2 and 
keep 1? 

What number must I double to get 4? 

What number is one less than 5? 

What is 1-2 of 4? 

What is 14 of the number 1 less than 5? 

What number is one less than 4 of 4? 

Augusta had 5 cents. She lost 1 of them, 
and spent % of what she had left, and then 
found 3 cents. How many had she then? 
(Solve one step at a time.) 

49. For the method of making series like the 
following, see preface. For method of use see — 
pages 42 and 43. 

For addition.* 


* Teach each series thoroughly before taking the next. 


FIRST STEPS AMONG FIGURES. » P38 


a b Cc d e 
dag tt ce tates 
aL se Nh 4 1 2 

For subtraction. 

a b Ps d e f 2 
Bee > 3 54/3 1163 4 
Pees | 2 22 {2 1 [3.4 1 

For multiplication. 
a b Cc d e f g 
Riinleze Oblate (iy 2. Ort 9 4 
TeeetOn te u200'| 12. }0 <1] 2.0 
For division. 
a ae Cc d aes 
6 2 ear OG: Tra eo 
Lee: : PSE ee 
For addition. (Re-arranged.) 
a b £ d e x 8 h 
4 3 2113.0 Z 2 ve 
eeereisert | 3. 2) | 1; 3 2 : 2 
For subtraction. Bese ed 
a b Cc d e f g | h 
5 6 hy SE ee 
Baths 2 2 : 2 Ze sel 


50. Teach Roman notation to XVIII. 

Review counting by 2’s commencing with 2, 
and also with 1, to 60 and 61. | 

Teach counting by 3’s commencing with 3 
to 30. 


3 


34 FIRST STEPS AMONG FIGURES, 


The examples given under previous numbers 
should be frequently reviewed, so that the pupil 
may become quite familiar with the formation 
and use of numbers. 


SEVEN. 
51. At this stage pupils should make the 
schedule from their memory on the plan of 
those already given. 


Schedule : 


Measuring by tr. 
pie Ges ME? eee Fah a Gp 
1 hes Rens wit cet! ce 9 Weg as Bis NY domme 
Bap Gk 
7—I—1—I—1—1—1=1, or 7—1—1—1 
I—I—I—I=0, 


7=1=7 
By 2. 
II. 2. -2+2--2+ 107, 
L-E 2.) 2x34 1==7; 
II 2 7—2—2—2==1. 
1. 


7-9.. 7--2=3 (1 rem.) 


Teach pupils to write Arabic to 199. 


* FIRST STEPS AMONG FIGURES. 


Pera) 
Peetet 3. 3X 2+ Tae, 
I I 


Oe ea Gate B 


By 5. 


Pepreeie TE 6 5 2=7, 
It 2 5XI+2==7. 


Pelee tet tf 7.) >7—coez, 
7-5==I (2 rem.) 


ITri¢i31: 6. 6+1=%. 
It 6X1+1==9. 


Seer let Io 727 7 —6==1. 

7--6=1 (1 rem.) 
52. 
5ti=? 7—5=? 6+-4=—=? 
6—2=? IX5+I=? 47—3=—? 


35 


4 


36 FIRST STEPS AMONG FIGURES. 


SI=? 3X2+I—? 2X24+2—? 
2X2—? 6—2—2=? 7—2—2—2em 


3t+2=? 2X2? IX5+2=—? 
4—2=? 7+4=? 5—I=? 
6+1=? 3X2=—? 73 


2xX2+1=?2xX2+3=? 

53- 7 1S 3 more than? 

What is 4 of the number 1 less than 7? 

7 is 1 more than twice what number? 

What number is 3 less than 7? 

What number must be added to 3 to get 7? 

How many times can you subtract 2 from 7? 

7is 4+? 3 is 4 of? 

I added 3 to a certain number and got 5; 
what number was it? 

I pear is what part of 7 pears? 

2 pears are how many times 1 pear? 

How much must be taken from 7 to leave 3? 

What number must be added to z to get 7? 

How many times 3 must I add to 1 to get 7? 

Count by 3’s from 3 to 42. Review the 
counting. 

How many times can you subtract 5 from 7 ? 

54. 61s double what number ? 3 is 2 less than? 

What number is one less than 7 ? 

What is % of the number 1 less than 7 ? 


FIRST STEPS AMONG FIGURES. 


55. For addition. 


a b Cc d e f 
314 | 203 | 120 | 312 | 0131 | 420 
123 | 412 | 312 | 342 | 3441 | 234 
For subtraction. 

a b Cc d e f 
456 | 277 | 615 | 454 | 353 | 2643 
432) |-143 | 412 | 312-| 341] 2312 


For multiplication. 


Bin O81 | 102 | 50 
ee ia 2 132 | Te 

For division. 
a hae Perc d e 
26 | o1 40 | 36 023 
Ree feret |) 21 |_ 32° 322 


Same re-arranged. * 
For addition. 


a bofiec d |e f 
241 | 302 | 413 | 021 | 302 | 4130 
123 | 412 | 341 | 231 | 234| 1234 | 
For subtraction. 
aaab- Db Cc d @ f 
435 | 256 | 534 | 634 | 741 | 6375 
AZ2 |-143'| 412 | 312 | 341 | 2312 
For multiplication. 
geebelec t djive f 
20 | 12 | 03] 12] 031 
12 | 12| 31] 23] 123 


* For review. 


1 ochayag Waleed Bikace lis tad as 2 


37 


nr 


38 FIRST STEPS AMONG FIGURES, 


For division. 


a 1 bc | didseseme 
32 | 60; 40 | 63 | 201 
(123) 223i 240aes 


56. After earning 3 cts., Fanny had § cts. ; 
how many had she before ? 

How many must you add to 3 to make 7? 

Nellie has 2 pencils, and Sarah has 1 more 
than Nellie ; how many have Jo¢/ of them ? 

Marcus has 4 marbles and Arthur has 2 less ; 
how many have both ? 

Mary has 6 pins and Stella has 2 ; how many 
more has Mary than Stella? 

57. When the pupils become listless or rest- 
less, or a minute or two of spare time is at com- 
mand, the following examples and like exam- 
ples given through the book will be found both 
useful and interesting. Use them often. 


34+242—-34+242—-1X2+ 37 2h ee 


Ans. 4. 

§—3 +1 X2+3+2—-34-4—2 3 2 
+1? Ans, 7. 

4+ 3+-2—-3+1%2—3++2-—8ix oe 
—2? Ans. 5. 


7—2—-344—243—5 +1 Ke 3oaeee 
—2? Ans. °. 


FIRST STEPS AMONG FIGURES. 39 


2+5—3--2—2+3X2—2+2x3+2-11 
ams. 2, 

ee ey 5 A 2-3 1X3 — 4+ 38+ 
4—47% Ans. 3. 

6— 44-3 +2—1-3 X2+3-—44+24+24+3+ 
eet! Ars 216. 

59. Teach the pupils to read numbers to root. 

Show them that the figure in the third place 

represents hundreds (or is named hundreds ) 


Read the following numbers : 


toe 2 2. 395. awe Uy AS i 
5: 341. 6.1704: y Pew AY Ds OLA, 
Ga 7oue 10. Sgt, Lkee Oo, 12, 260. 


Someta 14.) 750, 15. 981. LOs8 700; 
Teach the pupils that the fourth place rep- 
resents thousands (or is named thousands.) 
Teach the pupils to write a comma between 
hundreds and thousands before reading a num- 
ber. 
Read: 
BeOS eld. 374%) LO75410) 420. 7308. 
BeeOtoOen 227.7075.  2le Al18.. :.245:.9400. 
2eam7504.. 20. 1000, 27. 7050. -28. 8004. 
60. Teach pupils to write numbers to 1,000 in 
Arabic, and in Roman to XX, 


~ 37RTB 


40 FIRST STEPS AMONG FIGURES. 


1. Write two hundred forty-five (in Arabic.) 
2 “* five hundred sixty-one. 

3 ‘¢ one hundred thirty four. 

4 ‘¢ seven hundred twenty-one. 

5 ‘¢ three hundred eighty-six. 

6 ‘“¢ four hundred sixteen. 

7 ‘“¢ nine hundred twenty-one. 

8 “¢ six hundred thirty-two. 

9. ‘¢ eight hundred seventy-nine. 


61. Give more examples like the above until 
the pupils write them readily. Then give the fol- 


lowing : 

10. Write three hundred nine. * 

Pt: “eight hundred forty. 

Vip “‘ four hundred fifteen. 

r3: ‘* six hundred thirty-seven. 

14. “ two hundred ninety. 

1a “sixty-four. 

16. “« seven hundred two. 

ay Fe ‘“* five hundred six. 

18. “* three hundred thirty. 

IQ. ** four hundred one. 
Review these often. 
62. 


g+24+343 4+ 24+2434+3+3 245 ee 
Ans. 31. 


FIRST STEPS AMONG FIGURES. 4i 


Mae Stet oto tots F242 242 


pare acns, 27. 
mg poe t2t+3+3+3+3 F242? 
Ans. 28, 


oP HG oe i BPS eae ps Be Sd i SB 
Sogrias i PANS. 13. 

6+2—1X3—44+1X2+3+34+3+2+2 
+3437: Ans. 25. 

Vee aeie te 1 ot St 3+ 24 2 + 2-23 
3+2? Ans. 29. 

Sage 3t3t2t2t+2+2+3+3+3 + 2? 
Ans. 33. 

fee 3 2 tes +2 te 3 45 St 3 
+2+2% Ans. 25. 

Some boys are sliding down hill. There are 
3 sleds and two boys on each sled, how many 
boys are there ? 

James had 7 apples; he ate one and gave 
his sister half of the rest. How many did he 
give his sister ? 

John had two apples ; he cut each of them 
in halves. How many halves had he? 

How many horses in 2 two-horse teams ? 

A stingy boy had 5 sticks of candy; he 
would neither eat any nor give any away. How 
many did he keep? 


42 FIRST STEPS AMONG FIGURES. 


A generous boy had 3 sticks of candy; he 
gave his sister 2 sticks and he ate half a stick. 
How many had he left? 

Ralph had 5 peaches ; he gave 2 of them to 
his little sister, I to his father, I to his mother, 
and ate I himself. How many had he left ? 

64. Count by 3’s from 3 to 5o. 

Count by 2’s from 2 to 60. 
Count by 2’s from I to 61. =. 
Teach Roman notation to XXVIII. 

Write in letters.19, 13. 21, 14, 11, $,)8O,; 2m, 

12,:20,:177,01025, 27smeos 

66. Count by 3’s from I to 22. 

Review counting. 


EIGHT, 


67. Pupils may make the schedule like pre- 


vious ones. 
68. Give only a small part of these series 


each day and give with it slate examples from 
pp. 93 to 94, and oral exercises in the exam- 
ples following the series. By this variety much 
mo.e work may be accomplished without the 
weariness resulting from too much sameness. 
The series should on each succeeding day be 


FIRST STEPS AMONG FIGURES. 43 


ey 


reviewed.* For a review after completion take 
the re-arranged series. | 
For addition. 


a b c d e f g 

Bryne 135.2 4 3214135455 a AsY 
Meee a 120) 2-4 1 | 9273 T)}-2 2.434 1 234 

For subtraction. 

a b C d e f- | 2 
O5310350/8521/8571476)47.5|4 
feeeeteeteseii2 2.1 j.4 3-213 41) 2 3-4 1:3 

For multiplication. 

a b ¢ d e 
2 GARE OP TaD Ore Reg 
ee ier | 02" 2" 14? 3 

For division. 

a b fe d 
Meet tao 251-0. 3021.6 28 
Gemeente a Ios 122-2 354 

For addition. Re-arranged. 

a b é d Creole b g 
WA S144 2531042 153%) 425): 3 
Peres | AS 19.471) 2.3 Ail 3T. 236) 4 

Fcr subtraction. 

a b C fa gd fates 2 
345|627|665/484/875]567) 3 
213)213/412)342/3411432 1 


* Leave on the board the previous day’s lesson in series 
and add to it as much more ofthe series as can be mastered 
with the review. 


44 FIRST STEPS AMONG FIGURES, 


69. Teach Roman notation to XXX, 

What 2 equal numbers make 8 ? 

What is half of 8? 

What number is one less than half of 8? 

What number can you double and get 8 ? 

From what number can you take 2x3 and 
have I left? 

Count by 3’s from 1 to 4o. 

I write a number four times and add._ I get 
8, what is the number ? 

Henry had half of 8 cents. How many had 
he? 

8 cents will buy how many I-cent stamps ? 

4-cent stamps (I. Revenue) ? 2-cent stamps? 
5:cent stamps? 3-cent stamps? 

3 lemons is 3? of how many lemons ? 

Lewis brought 6 eggs from the barn ; he broke 
half of them. How many whole ones were 
left 2 

George has 3 cents, he finds 2 cents ; how 
many must he earn to have 8 cents ? 

William had 2 sticks of candy, he ate half of 
a stick, and his sister half a stick ; how much 
candy had he left? 

What number is I less than half of 6 ? 

Jane’s bird hatched 3 young birds and there 


FIRST STI PS AMONG FIGURES. - 45 


were 2 eggs which did not hatch; how many 
eggs in the nest at first ? 
What is half of the number I less than 7 ? 
William bought 3 marbles at 2 cents each ; 
how much should he pay for them? 
How many tops at 3 cents apiece can Ed- 
ward buy for 8 cents? For 7 cents ? 
6 cents is 4 cents more than Robert’s money, 
how much money has he ? 
3X2—4+3—2X1—1X3+1=? Ans. 7. 
7—3+2+2—1%X4—3 is how many less 
than 8? Ans. 3. 
2+3—I+2—1X6+2+5—2=3 is + of 
what number? Ans. 4. 
6+2—5+1+2+1x2. is 1 less. than? 
TIS; 7 
5+2—-153X4724+34+3+3+3+3+2+ 
Pia AUS, 23. | 
3723 3-1X34+3+3+3+3+3+2+2+ 
ef ANS. 25, 
j—a—teavqtatitgigt2tatet 
. at a2 ANS. 3%. 
§+3> 4x3¢+3t3+3+3t2+2t+3+1 
+3+2% Ans, 31, 
Seton L 2 2st s+3+2+2+ 3+ gt 
34243? Ans. 29. 


46 - FIRST STEPS AMONG THIGUEES. 


Seer Wer a + 3 aa 3a 
Ans. 27. 

44+3+24+24+34+34+3+24+24+34+3+2+ 
S72 Fv ANS, 24. 

643 +2 ieea-eess eae pate sac 
ANSie3 3; 

44+2434+34+3+2+2434+3+2+2? 
Ans. 29. 

7+34+-34¢34+3+24243 73550 
Ans. 30: 

442434343 4.224223 ee 
3-133? SAnseetos 

6+3+2+2—3 X% 24343735 34 5eeee 
2t2+2? Ans. 27. 

Oral exercise. Place the figures of any of 
these examples which involve only addition in 
a column on the board, let one pupil add them 
upward. then another downward, then another 
add them upward, but commence with I and so 
get a result I greater than before, add down- 
ward in the same way commencing with one ; 
then use 2 instead of I and also 3 instead of 2. 
This makes 8 different examples instead of one 
and gives excellent practice. ; 

How many feet have 2 pigs ? 

8 is 2 more than what number ? 


FIRST STEPS AMONG FIGURES. 47 


8 is 2 more than twice what number ? 

How many legs have 3 hens ? 

What number is 5 less than 8? 

How many times can you substract 3 from 8? 

How many legs have a rabbit and a bird to- 
gether P 

8 peaches are how many times 2 peaches P 

1 pencil is what part of 8 pencils? 

How many wheels have 4 sulkies ? 

What number must I add to 5 to get 8? 

4 is half of what number ? 

From what number can you take 2 times 2 
and have 3 left ? 

2 is how many less than 6? 

Count by 2’s from 1 to 62. 

6é 6“ 3's iT4 3 to 60. 
Emeeass, ‘4 2 t0.60; 
ame o3'sy oS P ta Gt, 

Teach Roman notation to XXXIX. _ 

Write in letters 26, 18, 34, 9, 16, 22, 37, 19, 
35) 21, 36, 39, 17. 

7 is how many more than 5? 

John had 7 sticks of candy; he ate 3 and 
gave away % of the rest. How many did he 
give away? How many did he keep? 

Count by 3’s from 2 to 62. | 


48 FIRST STEPS AMONG FIGURES. 


Teach Roman notation to L. 

Write in letters 34, 16, 25, 10,540, 20m 
17, 26, 33, 46, 37, 22, 50, 44, 27, 33, 49. 

Write in figures XVIII, XXXIV, XV, XL, 
XXVI, XII, XXXVI, XIV, XVI 


NINE. 
Pupils make a schedule. 
For addition. 
aoe. ab Cc ps 24 yet ah | g 
41513241153 at 136/241 |362 
12314121345 31412|345|234 
h i j 
451/532 
511] 235 
For subtraction. 
a b c d e Bad 
736 | 498 | 756 | 948 | 585 | 384 | 757 
412 | 345 | 125 | 312 | 341 | 232 | 345 
viene ha Kk cas 
672 | 69 
327 tei 
For multiplication. 
a={ bel vce edie esl ee ee h i 
24. | 101] 21)30)21 | 30 |s2a ae 
{25/231 (521401 ht eel sarees 234 


FIRST STEPS AMONG FIGURES. 49 


For division. 


wears: |.d- psecpf fogs | oh i 
30 | 44 | 30 | 61 | 84 | 94 | 42 | 062 
Papieea tie.) 31) 22, | g2 |.¥2 |321 
Re-arranged and including o. 

For addition. 

a b c d e f g h 
526 | 301 | 452 | 831 | 740 | 523 | 214 | 365 
123 | 435 | 123 | 051 | 226 | 341 | 723 | 614 

I | 
2201014 
529 | 34 
For subtraction. 

a b Cc d e f go. fy h 
268 | 643 | 485 | 756 | 796 | 467 | 967 | 759 
123 | 412 | 453 | 215 | 430 | 213 | 465 | 123 

i k l 

835 | 6839 

404 | 3215 
For multiplication. | 
Serepeec. |) die |e fojig. | h pk 
Aaeta20 (si. | 20 (31 | 24 | 14-| 20 
Popieaz. | 23.| 12°] 34 |-22°) 11} 14 
For division. 
epimDatecet d)|.e.|)f Jogo) hf 
32 | 09 | 40 | 61 | 86] 04 | 82 | 343 
Geer su 4F 2 ties | ar | 42°} 125 


4 


5° FIRST STEPS AMONG FIGURES. 


How many 1’s make 9? 

From what number can I take 9g 1’s and 
have nothing left ? 

How many 3 cent stamps can you buy for 9 
cents? 2 cent stamps? 1 cent? 5 cent? 
4 cent? 

How many oranges at 9g cts. each can you 
buy for g cts. 2 

How many times can I take 2 from g and 
have 1 left? 

George had 9g peaches ; he ate one and gave 
you half of what were left, how many did he 
give you 2 

What number taken 3 times will make 9 ? 

Henry gave each of his 3 playmates 3 plums. 
How many did he give away? 

What is half of the number 1 less than 7 ? 

William has 9 wheels. He has wheels for 
how many three-wheeled velocipedes ? 

5 and how many make 9g ? 

What number taken twice and 3 added 
makes 9? 

What is half the number 1 less than g ? 

Theodore had g marbles, he lost 4 of them, 
how many had he left ? 

44-35 f=¢. 


FIRST STEPS AMONG FIGURES. §t 


8 is 3 and how many? 

1-3 of 6 and how many make 6? 

Count by 4’s from 4 to 20. Review. 

What number is 1 less than 4% of 8? 

g=6+? 
5 from g leaves? 
What cost 4 lemons at 2 cts. each ? 
What cost 2 marbles at 3 cts. each ? 
5+3—4—24+3+4-—6=? Ans. 3. 
34+4—2+4—34+3+2+1=? Ans. 4. 
64+3—5+4-—1X2+7+2=? Ans. 9. 
oo = Sree eg r4 e455 St 
Ans. 8. | 

Meera Sy 3K Sti 4 Pf Ant 3. 

Oe Ramer SP ws wn ld we SL 
20. 

st5r4tas et2t24343t2te+ 
34+3+2° Ans. 25. 

ee eo ee ee + 3+ 33 
+3+2+2—-4? Ans. 21. 

eee eto eo oot 
+2+3° Ans. 41. 

Beato see 332+ 3a 2 
+2+3? Ans. 39. 

eee o 7 2 e+ 24343434243 
es 2p Ans. 42. 


§2 FIRST STEPS AMONG FIGURES. 


3+242+-34- 2 £3 434S ree 


2. Maan toy 
St2t+5+3t2424:3 13 sage ees 
+2% Ans. 38. 


8+3+2+24+-34+2+3 + 32 eeeaee 
3-2 De ANS. 4e78 a 

44-:34243.53 5 34n2 aetna $244 
2-2! (Ans. i324; 

Review these often. 

Count by 2’s, commencing with 2 and with 
1, to 60 and 61. : 

Count by 3’s, commencing with 3, 2 and 1, 
to 51, 53 and 52. 

Count by 4's, commencing with 4, to 32 and 
back. 

James having 9 apples ate 1, and gave the 
rest to his sisters, giving them 2 each; how 
many sisters had he ? 

George had 9 oranges; he ate one of them ; 
if he were to give you half of what were left, 
how many. would you get ? 

Charles having 9g pears sold 3, and gave his 
sister half of what he had left ; how many did 
he give his sister? 

Teach Roman notation to LXX. 

Write in letters 67, 44, 36, 59, 62, 46, 28, 16, 


FIRST STEPS AMONG FIGURES. 53 


Write in figures XVII, LXV, LXVII, 
LXIV, LXIX, XXIV, XIX. 

Pupils read the following, which should be 
copied on the black board, and the pupil who 
reads any number to point off the periods him- 
self before reading. 

TeG205- 2.7020. 3.8005. 4.9400. 5.1641. 
6.6780. 7.5416. 8.8605. 9.5400. 10.7508. 
11. 4870. 12.5718. 13.5851. 14. 6504. 15.5790. 
19. 1432. 17.9007. 18. 5000. 19. 7400. 20. 8040. 
25.4037, 22. 5919. 23. 7990. ‘24. 7803. 25.7001. 

Teach Arabic notation to 10,000. 

Teach the pupils carefully, as being of the 
utmost importance, that they should place a 
comma after the number expressing thousands 
and before they write the units period. Teach 
them that units period takes three places, and 
show them that when the number does not fill 
the three places, the places on the //¢ must be 
filled with ciphers. 

The following are to be read by the teacher 
and the pupil is to write them in Arabic: Re- 
view these often. 

1.4,573 2+ 3,240 3.5,296 4.7,315 
5. 2,324 6.7,560 7.1,427 3. 3,670 9. 7,305 
10. 5,741 If, 2,816 12. 51980 Iz $407 1 4-4,300 


54 FIRST STEPS AMONG FIGURES. 


15. 9,700 16. 4.315 17.8,590 18. 1,731 
19. 7,800 20. 5.004 21. 3 060 22. 5,104 
28. 6,003 24. 8,600 25. 9,419 26. 8,040 
27.5,900 28. 4,307 29. 5.009 30.9,016 
31. 9,300 32. 9,070 33- 7,049 34- 3,900 
35- 7,008 36. 7,080 37. 5,700 38. 7.000 
39. 55075 40. 3,716 41. 9,060 32. 5,800 
43. 6,904 44. 6,008 45. 7,600 46. 8,009 

Cut an apple into 3 equal pieces and teach 
the pupils that we call one piece oné third. 
Break a stick of candy into 3 equal pieces and 
so illustrate the same thing. 

In the same way teach one-fourth by 4 divi- 
sions’ instead of 3; then one-fifth, one-sixth, 
&c.’ When the pupils are familiar with this 
show them that of any 3 equal things, one of 
those things is one-third; of 4 equal things, 
one of them is one-fourth. Do not leave this 
subject until the pupils are very ready with 
their answers to the following questions : 

An apple is cut in 5 pieces; what do we call 
1 piece’? | 

An apple is cut in 3 pieces ; what do we call 
Te pieces 

An apple is cut in 6 pieces ; what do we call 
I piece? 


FIRST STEPS AMONG FIGURES. sl 5 


An apple is cut in 10 pieces ; what do we call 
1 piece ? 

An apple is cut in 4 pieces ; what do we call 
I piece? 

One apple is what part of 7 anes 

One apple is what part of 9 apples ? 

One apple is what part of 6 apples? 

One orange is what part of 4 oranges? | 

One pencil is what part of 8 pencils? 

A boy having 5 apples gave away one of 
them ; what part of his apples did he give 
away? 

A little girl had 6 peaches ; she gave one- 
third of them to her brother. How many did 
she give him ? 

(Teach the pupils that they get one-third of 
a number by dividing by 3; 4 of a number by 
dividing by 4, &c.) 

A boy gave away one-fourth of his 4 mar- 
bles ; how many did he give away ? How many 
did he keep ? 

If I divide 6 apples equally among 3 boys, 
what part of them do [ give each boy? How 
many do I give each boy ? | 


TEN. 
Pupils make schedule. 


56 FIRST STEPS AMONG FIGURES. 


For addition. 


a1 bo jecr id 
536|247|503 247 
234|5021364 623 
For subtraction. 

ate Da Cyd 

1098/758|676/6107|1074'389 1085 

623|406|223/6 45] 532/054] 323 

For multiplication. 
aot Desc | d |e) f tg 
3521313|240/132 421/521] 20 
123/012 415|231|203|124| 53 
For division. 
b|c 
842/920,616 
412 313/213 
Re-arranged. 
For addition. 
ac DwECtiad 
635/264,635|207 
23415021345 1652 
For subtraction. 
a |b co [id sce tie heat i 


8410 839|796|869'5710/897/75917 105 86510 
52 31406|543|223| 


e!| f-|:o 20S taaee 
535/245 623,043/6234 
14501235|236|742|3405 


e re go. | hoieas 
983/5969 
643|/2345 


a 


ete 
1038 035 


212|231 


f 
44 | 
24 | 


e 
463 
345 


Fo seath 
257/430|524 
120314641235 


i 
3524 
2346 


54 § 623 235/0 42.345 6 
For multiplication. 
a | b-}-c°| d)|- ete t.|2o7| Sie 
42] 14:|-24 | 3£ |-2§ 4 02 | 31 |) oneness 
12 | 12] 30 | 12:|\.4r-|)367)0 237 eee 


FIRST STEPS AMONG FIGURES. 57 


For division. 


Meeiecc|.dj.¢| fg | h |i 
18 | 60 | 32 | 85 | 44 | 02 | 106 |} 310 | 94 
Searle.) At | 12 |31 | 22 | 3 5 1-34 


What is 3 of 10 buttons? 

Lucy had g pins; she lost 4 of them and 
then found 1 ; how many had she then? 

4 and how many make Io? 

10 boys were out in a sail boat; 1 more 
than half of them were drowned ; how many 
were drowned ? 

Minnie had 9 cents ; she spent 5 of them 
and lost 4 ; how many had she left ? 

10 cents will buy how many 3 cent stamps? 

How many 2 cent sticks of candy can you 
buy for to cents ? 

How many cents will 4 two-cent marbles 
cost ? 

Lucy had 5 cents and her mother gave her 
4 cents, how many cents had she then? 

Show the pupils that we call 2 and 2 equal 
numbers, that they are equal to each other, 
also 1 and 1 are equal numbers, 3 and 3, &c. ; 
2 and 1, or 2 and 3, or 1 and 3 are unequal 
numbers. 

What 2 equal numbers make 4? 

What 2 unequal numbers make 4? 


58 FIRST STEPS AMONG FIGURES, 


What equal numbers make 3? 

What unequal numbers make 3? 

What 2 equal and 1 unequal numbers make 
-§? (Two answers.) 

How many wheels have 5 sulkeys? 

How many wheels have 3 three-wheeled 
velocipedes ? 

Mary was bringing in 10 eggs in her apron, 
she broke 2 less than half of them; how 
many did she break ? 

How many were unbroken? 

John had a string ro yards long and William 
had one three yards long; how much longer 
was John’s string than William’s. 

6 and how many make 10? 

What number taken from 10 leaves ae 

What cost 5 two-cent stamps ? 

Henry had 10 miles to walk, he has walked 
4 of them ; how much farther has he to walk? 

1o beans are how many times 2 beans? 

8 boys were playing “snap the whip,” 6 of 
them kept hold of hands; how many were 
there that did not let go? ; 

What number added to 3 will make 10? . 

Count by 4’s, commencing with 4, to 60, 


FIRST STEPS AMONG FIGURES. 0) 
Review counting by 2’s and 3’s. 

For rapid solving. 
AF3Z—-UF2X3+ISHSX4+4+ 243 + 3 
te Gee ee eae ANS. 24. 
ee Ae 2X2 +3434 3.4342 

+24+34+3+4+4+4=? Ans. 4o. 
Bees ho aga ot 2s It 4 a 4 4 
+4+4=? Ans. 44. 
Sees 4444-47 3434-3 +2 +2 
332-3 ?. Ans. 48. 
mnene ho 2 2X 8+ et At Aa 4 +3 
pe 3 3. oe 2 Ans.36. 
fees O41 4 AT 43 2b 2 +3 
ee Gt Bae th asa Te 
Seas a 4 23 t3t34r2e +2 + 37+3 
+3+3+2+2+3+3=—7?. Ans. 38. 
40-3 Sao TSt3 t 3+ 2h 4447444 
ee h2b2+3+3—) Ans: 55. 
Poa at aes 2-2 2 X38 oo 3 2 2+ 
34+1+4+4+4=7? Ans. 29. 

Beeson 4 A AAT AAT 4 tr 3 +3 
+3+2+2+2+2=—7 Ans. 57. 
iweeres wot 2 Ad 4 4.4.3 4-3: 3412 
+2+3+4+4+3=2? Ans. 55. 
§+34+44+4+4 + 44+3+2+3+ 44848 

+4+8+2+8=—7 Ans. 74. 


bo FIRST STEPS AMONG FIGURES. 
4t+3t2ti1+4t+4+4+4+3+2434+3 
+2+34+3+3+3=? Ans. 51. 
For more practice see pp. 51 and 45. 
Count by 4’s from 2 to 34. 


(T9 (75 4’s 46 4 75 60. 
cc éc 3's 6c 3 ce 60. 
66 ce 3's cc I 73 6r. 
6c (79 3S 6¢ 2 (73 62. 
T3 (79 2’s cc 2 ce 60. 
ce Ce 2’s 6s I éc 60. 


Write in letters 36, 41, 16, 64, 56, 47, 69. 

Write in figures XLV, XV, LXIV, LXXXV, 
XXVII. 

What equal numbers will make 6? (3 ans.) 

What unequal numbers will make 6? 
(Several answers.) 

What 2 equal. numbers and 1 unequal 
number will make 6? 

The following numbers are to be copied on 
the blackboard and the pupils are to be re- 
quired to point them off in periods and read 
them. 

I. gI0or7. 2. 86700, . 3: 90007. ay faogae 

5. 70000. 6. 50010. 7. 38419. 8. 74058. 
9. 60800. 10. 16040. II. 3000. [2. 7014. 
13. 10061. 14.-- 3020. 15. : 7003: *165 6600: 
r7, 17500. 18. 3540. 19: 67374)020moggem 


FIRST STEPS AMONG FIGURES. bi 

What 2 equal numbers and 1 unequal num- 
ber will make 7? (3 answers.) 

What 3 equal numbers and 1 unequal num- 
ber will make 7? (2 answers.) 

What is 1-3 of 9? 

I is what part of 7? 

Teach Arabic notation to 99,000. 

Show the pupils that they should place a 


comma after the figures that express thousands 
before writing the unit period, and a period at 


the end of the number. 

The following numbers are to be read by 
the teacher and written upon the blackboard 
or slates by the pupils. 

beeeye300., 2) «53000, 3.°.2,050. 4. 10,091, 
Remos0102" 6; 4,000. 7. 12,090. 3. 50,700. 
OmGs,070. 010) 45,0132 11. $0.000. 12. 80.010. 
13. 15,061. 14. 40,002. 15. 79.500. 16. 81,018. 
17. 30,600. 18. 60.060. 19. 90,004. 20. 8,050. 
G7 5.060, 22, 74.695. 23.-31,280. 24. 13.300. 
25. 14 O41. 26. 10,010. 

Teach Roman notation to C. 

Write in Roman 1. 64. 2. 49. 3.97. 4. 76. 

Write in Arabic 5. XIX. 6. LXXXIV. 7. 
Doese VIL, 9. XVIII. 10, XX1X. 

Write in Roman 11. 17. 12. 56. 13. 83. 14. 


49. 


62 FIRST STEPS AMONG FIGURES. 

Write in Arabic, 15. LX VID Gee ee 

If the teacher prefers it, the pupils can buy 
their books (the Pupils’ Edition) at this stage, 
and do more slate work than the teacher could 
have time to dictate to them, or copy upon the 
board for the pupils to copy and solve. If the 
pupils do not have their books the teacher will 
assign daily lessons from page 1, 2, &c., of the 
Pupils’ Edition, doing it in connection with this 
work and thus carrying on that work together 
with the following work. 

The teacher should now begin to give the par- 
allel work of the Pupils’ Edition in connection 
with that of the Teachers Edition. The parallel 
pages are denoted by the numbers at the bottom 
of the pages in each. 


FIRST STEPS AMONG FIGURES. 63 


se aa ae Ne 


Counting two or more numbers into one 
number is called Adding, or Addition. 

The number obtained by counting two or 
more numbers into one number is called the 
sum .of those numbers. 


For addition and multiplication. (5 and rev.) 


a b cf speadap hse f g 
Be to 4 | P35. 4401 35.2.) 4 1 
ee oe | 452-1345 | 234 1.523 £4 5 
For subtraction. 

a b Cc d e ii g 
7410 | 746 | 758 | 693 | 696 | 857 | 8 5 
Paes fo | to | 345 1432 [5 “4 
For division. 

a b Cc d e 
41s10} 6165 | 20 g2 | 20815 | 610 8 
Seem imontont 43201) 154.3 ||) 255? 

f g 
Breen 4 t2 
2) AeA Bea) 


See Pupils’ Edition, p. 5. 


Og FIRST STEPS AMONG FIGURES. 


It is well to require pupils to bring a written 
analysis of an example to recitation and to 
give the solutions of other examples orally in 
class in the same form, but there should be a 
large number of examples given, of which ond 
the answer is to be given and that as soon as 
possible after the reading. 

1. Susan had 4 cents and her mother gave 
her 3 more ; how many had she then? 

Solution. She had the sum of 4 cents and 
3 cents, or 7 cents. 

2. John has 5 marbles and James has 4 mar- 
bles ; how many have both? 

3. Lulu has 3 eggs in one hand and 2 in the 
other ; how many has she in both? 

4. Walter bought some candy for 4 cents, 
and some raisins for 5 cents; how many cents 
did he spend? | 

5. Martha read 4 pages in the forenoon and 
2 in the afternoon ; how many did she read that 
day? 

6. A boy had 3 pencils in one pocket and 5 
in another ; how many had he in both? 

7. If a top cost 4 cents and a marble cost 2 
cents, how many cents must a boy have to buy 
a top and a marble ? 


FIRST STEPS AMONG FIGURES. 65 


8. Jane bought 2 books; she had 3 before. 
How many has she now? 

g. Henry walked 4 miles before dinner, and 
4 after dinner ; how far did he walk that day? 

to. There are 3 barrels of apples under one 
tree, and two under another ; how many under 
both? 

Ask the pupils to bring examples of their 
own to recitation different from those given 
them. The teacher also will make additional 
examples, using pleasant facts about the school 
room, or the pupils, or their homes, something 
they have seen. 

Count by 4’s from 2 to 62. 

Taking one number from another number is 
called subtracting,,or Subtraction. 

The number obtained by taking one number 
from another number is called the Remainder 
or Difference. 

1. Joseph had 8 cents; he spent 5 cents for 
an orange. How many cents had he left ? 

Solution: He had left the difference be- 
tween 8 cents and 5 cents, or 3 cents. 

2. Mary had a cake which she cut into 10 
pieces; 7 were eaten. How many were left? 

3. My knife has 6 blades; 2 of them are 
open. How many are closed? 

See P, Ed., p. 9. 


b6 FIRST STEPS AMONG FIGURES. 


4. James bought a paper for 5 cents ; he gave 
the newsboy 1o cents. How much change 
should James receive ? 

5. Samuel put 9 peaches on the table and 
his sister took 5 of them ; how many were left ? 

6. A man owed $7 ; he paid $3 ; how many 
$’s did he then owe? 

7. Willis took 9 cents to buy candy with ; he 
lost 4 cents. How many had he to buy candy 
with? . 

8. Henry bought a pencil for 4 cents and 
sold it for 7 cents; how many cents did he 
gain? 

8. Matthew bought one pencil for 4 cents 
and another for 5 cents; what did both cost 
him ? | 

to. Susan bought 3 spools of white thread 
and 6 spools of blue thread ; how many spools 
did she buy? 

11. Jane tried to solve 6 examples ; she had 
4 of them correct. How many were wrong? 

12. Walter had 8 pencils ; he broke 3 of 
them. How many whole ones had he? 

13. Fanny had 6 needles ; she found 4 more. 
How many had she then? 

14. May is 7 years old and her brother 


FIRST STEPS AMONG FiGURES. 67 


Frank is 4 years old ; how much older is May 
than Frank? 

15. George had a stick g inches long ; he cut 
off 3 inches of it. How long was the stick then? 

16. A farmer having 8 turkeys, sold 4 of 
them ; how many had he left? 

17. John paid 3 cents for candy and 5 cents 
for marbles ; how many cents did he spend? 

18. A little boy had 3 fingers cut off ina 
machine ; how many had he left ? 

1g. Silas had 3 marbles in one pocket and § 
in the other ; how many had he in both? 

20. How many wheels have a sulky and a 
wagon together ? 

21. There are 3 girls on the front seat of a 
carriage, and 5 girls on the back seat ; how 
many girls in the carriage ? 

22. Jesse had 8 sticks of wood to bring in ; 
he has brought in 3 ; how many more has he to 
bring in? 

23. An orange cost 6 cents, and a peach cost 
3 cents; how much more did the orange cost 
than the peach ? 

24. If a pear cost 4 cents, and a lemon cost 
5 cents, what will a pear and a lemon cost? 

25. There were 6 eggs in a nest and 4 of 

See P, Ed., p. II. 


68 FIRST STEPS AMONG FIGURES. 


them were broken ; how many whole ones were 
there ? ; 

26. Ellen’s father gave her g cents ; she 
bought a doll with 5 cents. How many cents 
had she left ? 

27. There are 4 boys riding in a sleigh and 
2 riding behind on the runners ; how many 
boys with the sleigh ? 

28. Ella has 5 roses on her bush, and 5 in 
her hand ; how many has she? 

29. There were g chickeas in a coop and a 
rat ate 3 of them; how many were left ? 

30. There are in the class 4 girls and 3 boys ; 
how many pupils in the class? 

31. A little boy bought 10 sticks of candy ; 
he ate 3 of them and gave away the rest. How 
many did he give away ? 

These 31 examples should be reviewed and 
others given, until the pupils know at once in 
such simple problems whether they are to find 
the sum or the difference. Review the series 
also. 

Count by 4’s from 1 to 17. 

Copy the following examples one at a time 
on the blackboard ; require a pupil to point one 
off into periods and read it. Erase it, then 


FIRST STEPS AMONG FIGURES. 69 


write another and require another pupil to point 
it off and read and so with the others: 


I. goo16. 2. 45378. 3 461340. 
4. 908714. 5. 876341. 6. 608790. 
7. 379000. 8. 75608. Q. 40713. 
10. 100740. It. 98716. I2. 900000. 
13. 800601. 14. 200003. 15. 761300. 
16. 500000. 17. 700300. 18. 60050. 
19. 700060. 20. 600000. 21. 200361. 
22. 500700. 23. 40010. 24 900007. 


Teach Arabic notation to 999,000. 

To be read by the teacher for pupil to write 
upon slates or blackboard. 

Write in Arabic the following : 


I. 1,040. 2. 3,506. 3. 10,016, 
4. 8,400. 5- 9,350. 6. 7,518. 

aa Lye 8. 10,010. 9. 40,070. 
ore7 5-501." 11. 36,000. 12. 90,090. 
13. 100,100. I4. 702,940. 15. 900,070, 
16. 816,902. 17. 49,049. 18. 860,705. 


5 407,017" 20. 791,486.21. 21. 10,020. 


Write these, or similar numbers on the board 
and require the pupils to read them. 
Write in Roman the following: 
ta 0: 2. 96. 3. TIO. 


4. 47. Sori: 6. 134. 
See P. Ed., p. 26, 


56 | FIRST STEPS AMONG FIGURES. 
_ Write in Arabic: 
Walia a. oe be g. CXC. 
10° CXLIV. 11. :\CLXXXV Ista sais 
Write in Roman: 
13.0192... ¥4. 136. 915..1000 Beaten 
A short method of adding equal numbers is 
called Multiplication 5 
or, 
Taking a number a certain number of times is 
called Multiplication. 
The number obtained by multiplication is 


called the Product. 

1. John bought 5 pencils at 4 cents each; 
what did they cost? 

Solution: They cost 5 times 4 cents, or 20 
cents ; or if one pencil cost 4 cents, ve pencils 
will cost 5 times 4 cents, or 20 cents. 

2. If 1 orange cost 5 cents, what will 4 
oranges cost ? 

3. What cost 4 marbles at 3 cents each ? 

4. How many quarts in 2 gallons? 

Solution: In one gallon there are 4 quarts, in 


2 gallons there are 2 times 4 quarts, or 8 quarts. 
5. How many pints in 3 quarts ? 


6. How many quarts in 5 gallons? 
7. Ilow many wheels have 3 wagons? 


FIRST STEPS AMONG FIGURES. 71 


8. A lady gave 3 little girls 5 bunches of 
grapes each ; how many bunches did she give 
them all ? 

9. How many feet have 4 hens. 

to. How many feet have 3 cows? 

11. What cost 5 books at 4 shillings each ? 

~12. What cost 3 lead pencils at 5 cents each ” 

13. If a lead pencil cost 6 cents and a mar- 
ble cost 3 cents, what will both cost ? 

14. What cost a doll worth 4 cents, and a 
spool of thread worth 6 cents ? 

15. What cost 3 pencils at 4 cents each ? 

16. The boys are riding down hill on sleds ; > 
there are 4 sleds and 2 boys on each sled ; 
how many are riding down hill ? 

17. ‘A boy bought a sled for 8 shillings and 
sold it for 5 shillings ; how many shillings did 
he lose ? 

18. A boy had 8 cents in his pocket, but he 
lost 4 of them through a hole in his pocket ; 
how many had he left ? 

19. A boy paid 4 cents for candy and 5 cents 
for nuts; how much money did he spend? 

Review these examples carefully. 

Count by 4’s from 1 to 29. 

For rapid solving. 

See P. Ed., p. 19. 


72 


FIRST STEPS AMONG FIGURES, 


12=- 4434313444444 4 ees 
+2+2=? Arnis. 41. 
34+44+413+313 +3 +24 25) coe 
+2+3+4=? Ans. 44 
3+3+2+2+2+3 +3+3 +293 tee 
+3+3+3=? Ans. 39 - 
34+44+41+4+3 +3 | 2 +27 
+2+2=? Ans. 41. 
24343734225 2er ies tees 
+3+4=? Ans.-37- 
3+2+2124+343+ 24+ 2 +1 3a 
+3+-7— 2 Ans. 22 
4+3+4+4— 3+4—-3—-2 + 444-342 
+3 ARS de 
24+-34744444—-342-— 3 — 260.00 
“312 SANSA TFs 
2+4+3+4+3 —4—-3 —4+3+31+414 
+3— ? Ans, 22, 
5+3—-2—-44+3+ 3 +24443144+3—-2 
—3=? Ans. 19. 
314724 45-24-2— fe teehee | 
+4=? Ans. 13. s 
g+2+4-3-2+ 3.434343 fae 
—4+3+2=2 Ans. 16. 
4+3+24+3+3+4+ 3—4-4+1—-34+4 
—3—4—4=? Ans. 5. 


FIRST STEPS AMONG FIGURES. 73 


20. What cost 5 marbles at 4 cents each ? 

21. What cost a marble at 3 cents and a 
pencil at 4 cents ? 

22. James bought 3 books at 4 shillings 
each ; what did they cost him? 

23. David had 8 apples when he started for 
school, but he ate 3 on the way; how many 
had he when he got to school ? 

24. Sarah ate 3 crackers at breakfast and 5 
at dinner; how many did she eat at both 
meals? 

25. How many horses in 3 four-horse teams? 

Finding how many times one number is 
contained in another is called Division. 

The number which shows how many times it 
is contained is called a Quotient. 

1. How many pears at 2 cents each can be 
bought for 8 cents? 

Solution: If 1 pear cost 2 cents, for 8 
cents you can buy as many pears as 2 is con- 
tained times in 8, or 4; or, aS many as there 
are 2’s in 8, or 4. 

2. John has 15 cents ; how many marbles at 
3 cents each can he buy? 
3. Willis spent 20 cents for oranges at 5 
cents each; how many oranges did he buy ? 
see P. Ed., p..25- 


74 FIRST STEPS AMONG FIGURE: 


4. If one pig cost $4, how many pigs may 
be bought for $12? 

5: How many lead pencils at 4 cents each 
can be bought for 20 cents? 

6. If 1 doll cost 3 shillings, how many 
such dolls can be bought for r2 shillings ? 

7. to shillings will buy how many knives 
at 2 shillings each ? 

8. How many balls at 3 shillings each may 
be bought for 6 shillings ? 

g. When pears are 2 cents each, how many 
can you buy for 8 cents? 

Count by 4’s from 1 to 61. 

6“ 6“ 4’S 66 2 & 62. 
6é 66 4’s 66 4 “ 60, 
3 6c 4’s 6 3 6“ 63. 

ro. What cost 4 pineapples at 2 shillings 
each ? 

11. How many pencils at 4 cents each can 
be bought for 16 cents? 

12. A boy walked 5 miles 1 day, and 3 miles 
the next day; how far did he walk in the 2 
days? 

13. How many pairs of mittens at 3 shillings 
a pair can you buy for g shillings ? 

14. 20 shillings will buy how many purses 
at 5 shillings each? 


FIRST STEPS AMONG FIGURES. 75 


15. Henry earned g cents on Monday and 5 
cents on Tuesday ; how many cents more did 
he earn on Monday than on Tuesday ? 

16. Louisa had 5 cents and she found 3 
cents more ; how many cents had she then? 

17. If one sied cost 5 shillings how many 
sleds can you buy for 20 shillings? 

‘8. What cost 4 vests at $8 each? © 

ig. How many neckties at 3 shilings each 
can you buy for 12 shillings ? 

20. 5 little boys each have a pair of copper- 
toed boots ; how many boots have they ? 

Review these examples. 
For addition and multiplication. (6 and rev.) 


a b c d e 
Poneeeael 1 362 (415-4 [0241 eee 
2345163451623415623 | 2456 

f g h 

Betetyy) 362.143 1'5 

2345|623/456 

For subtraction. 

a b | Cc | d 
fegoaro| 7 11 -7-8.| 3210 8 6.\-4 7.9.8 
SiO re4 | 5.15 4.3 1-2: 6-61) |-3 12.4.5 

e f g h 
Peereyets 080.5 6 | 9 10°5-| 5.9 6 
O45 432).6 541234 


ce Ls, pies, 


76 


For division. 


FIRST STEPS AMONG FIGURES. 


a 
80 127584 
52 6n253 
d 
AVi2gVaees 
2.3.4.5 
(G)- (2) 
32 33 
13 21 
3 2 
21 30 
13 13 
33 32 
LIS 131 
(7) (8) 
Oo 23 
23 33 
33 of 
12 22 
31 31 
23 33 
33 22 
185 194 


b c 
10-20 15 36 | a 2erceeeee 
2:4 § 6 | *4aieeeG 
e f 
18 12°6 16 175 @osgeTe 
623 415 62 3 
g h 
8 20 6/9 24 I0 
4, 5.9:1.30 eis 
(3) (4) (5) a6) 
2 8 33 Jt 
oe 33 3! 23 
23 3t 23 33 
33 I2 13 23 
21 22 22 32 
23 31 32 13 
134 152 154 155 
(9) (10) (xx) (x2) 
fed 3 3t ee 
a 23 33 32 
23 ot 23 +S 
3 23 39 33 
Ba 33 22 21 
23 of St 32 
33 oY 33 23 
75 166 203 177 


See P. Ed., p. 16. 


od 


FIRST STEPS AMONG FIGURES. iy 


13. Add 32, 21, 3, 33, 23, 31, 23, 33- 
14. Add 33, 23, 30, 21, 2, 33, 23, 22. 
Meme ln, 34922, 31, 23..32, 12;-23. 
Merrodest, 20, 22, 33°21) 32, 22, 32. 
17. Add 23, 13, 32, 23, 3, 13, 32, 33- 
Count by 5’s from 5 to 60. 

For rapid solving. 


374744343 42144+3+3 +2+4+4 
+3? Ans. 42. 
Pepe 4a oT oT OE L834 4+ 4+ 
Bees 20) ANS..ST. 
statatststatatstststat2t 
4+4+4¢ Ans. 57. 
eee 3 toe eg + 4+2+443 
+-2+3+5?. Ans. 46, 
Rages oot 43 3 t 2 FA 3 +4 
+3+3+4? Ans. 50. 
4+3+4+3—2—-2+5—3+2+4? Ans, 
18. 
2+453X5—4X% 4-3-3434 6+4X6 
+4+3—2? Ans. 23. 
3+64+44+44+4+34+4—-6—4—2—-5+3 
+4-—4? Ans. 14. 
34+5>2X5—3+4—-2—47+5 x6+4+3 
so53+31 Ans. ft. 


78 FIRST S11 PS AMONG FIGURES. 


BR cote ea X54 ee iatreg 
+5+2+4+37 Ansiaeee 
3+6+4—1 — 3x6—3—4-3 +4=6x5 
4 Pre 4g ei Seecns 
§5+6—3444+3413 +3 +4731+474-2 
—3+4-P3'-4 5 9 Ase e 
Do not use all of these at once, but use them 
occasionally and in connection with a lesson 
in examples of another kind, or to wake up 
the whole school» sometimes when they are 
listless. 
1. What cost a pencil at 6 cents and a mar- 
ble at 4 cents ? 
2. Mary had 5 peaches and her brother gave 
her 6 more; how many had she then ? 
: What Egat 5 books at 6 shillings each ? 
4. 3 boats are on the lake; each has a pair 
of oars, how many oars have the 3 boats? 
5. There are 9 bovs in a class, and 6 of them 
recite well ; how many do not recite well? 
6. How many baskets, at 4 cents each, can 
be bought for 24 cents? 
7. Charles spent 18 cents for candy at three 
cents an ounce ; how many ounces did he buy ? 
8. Jane had 10 needles, she lost 7 of them ; 
how many had she then? 


FIRST STEPS AMONG FIGURES. 79 

g. A lazy boy brought his mother 3 sticks of 
wood at one time and 4 at another ; how many 
sticks did he bring her ? 

Io. Six boys can sit on this seat, how many 
boys can sit on four such seats? 

ir. In a school room there are 6 keys hang- 
ing on a nail, 2 keys for each door ; how many 
doors are there? 

12. There are 18 words in the sp:lling les- 
son, 3 words for each pupil ; how many pupils 
in the class? 

13 There are 5 piles of books and 6 books 
in each pile ; how many books in the 5 piles? 

14. One stormy day George cleared the path 
of snow 4 times in the forenoon and 5 times in 
the afternoon ; how many times did he clear 
the path? 

'15. Amelia had 11 cents and spent 5 of 
them; how many had she left ? 

16. Arthur had 7 buttons on his jacket; how 
many had he after losing 2 of them ? 

17. How many quarts in 6 gallons? 

18. Mr. Smith has a quart of maple syrup; 
how many times can he fill a pint cup with it? 

19. How many skates at 6 shillings each can 
you buy for 24 shillings? 


80 FIRST STEPS AMONG FIGURES. 

20. What cost a knife at 5 shillings and a 
saw at 6 shillings ? 

21. If 2 oranges cost 12 cents, what will 1 
orange cost ? 

Solution: If 2 oranges cost 12 cents, I 
orange will cost + of 12 cents, or 6 cents; or 
1 orange will cost + of 12 cents, or 6 cents. 

Before giving examples like the above teach 
the pupils carefully that if 2 things of equal 
value cost a certain sum, 1 of them will cost 4 
of that sum ; if 3 cost that sum, 1 of them will 
cost 1-3 of it; if 5 cost any sum, 1 of them 
will cost 1-5 of it; if 9 of them cost any sum, 
1 of them will cost 1-9 of it, &c. 


Question on this subject until it is thorough- 
ly mastered. 


Show the pupils that to get 4 of 12 apples 
(or marbles or pencils) they may be placed in 
2 equal piles, and they will find that 3 of 12 
apples is.6 apples. 

Show them that to get.4 of 12 pencils, they 
may be placed in 3 piles, and that 4 of 12 is 4. 
Show in the same way that + of 12 is 3. When 
this is thoroughly understood, show them that 3 
of 12 may be obtained by dividing 12 by 2— 
the result in each case being 6; show that 4 


FIRST STEPS AMONG FIGURES. 81 


of 12 may be obtained by dividing 12 by 3; 
teach 4 of 12 in like manner. Illustrate also 
by + of 6 and 3 of 6 and ¢ of 8. 

Then teach in general terms that 4 of any 
number may be obtained by dividing the num- 
ber by 2; +, by dividing by 3; 7, by 7, &c. 

22. What cost 1 pear if 4 pears cost 8 cents? 

23. If 3 knives cost 15 shillings what will 1 
knife cost ? 

24. If 2 pencils cost 16 cents, what will 1 
pencil cost ? 

25. If 4 stools Fie 12 legs, how many legs 
will 1 stool have? 

26. If 6 boys earn 18 cents, how many does 
1 boy earn? 

27. If 5 cords of wood cost $25, what will 1 
cord cost? 

28. How many pounds in 1 box of honey if . 
4 boxes contain 24 pounds? 

29. At 5 cents each, how many oranges can 
be bought for 30 cents? | 

30. If 3 lemons cost 18 cents, what costs :1 
lemon ? 

31. How many pounds of butter will last a 
family 1 week if they use 12 pounds in 4 weeks? 

32. How many days will 18 apples last a 
boy who eats 3 apples each day ? 6 


82 FIRST STEPS AMONG F GURES. 


33. A blacksmith shod 5 horses each day ; 
how many did he shoe in 6 days ? 

34. 6 boys are skating on the ice, and 4 
boys are sliding on the ice without skates ; how 
many boys on the ice ? 

35. Nine boys were riding down hill on sleds ; 
3 of them went home. How many continued 
to ride down hill? 

36. If 15 yards of cloth will make 5 pairs 
of trowsers, how many yards will it take to 
make 1 pair of trowsers ? 

37. George has 4 books, and Mary has 5 
books ; how many have both ? 

38. 12 cents will buy how many marbles at 
3 cents each ? 

39. If 5 marbles cost 10 cents, when will 1 
marble cost ? 

40. What cost 1 apple if 5 apples cost 10 
cents? 

. How many dolls at 4 shillings can you 
Sg ae zo shillings ? 

Review the last 21 examples until the pupils 
solve them readily and can distinguish when 
they divide and when get one-half or one-third, 
&c. 

Count by 5’s from 1 to 61. 


FIRST STEPS AMONG FIGURES. 83 


Count by 4’s from 1 to 61. 
Count by 4’s from 2 to 62. 
Count by 3’s from 3 to 60. 
Count by 3’s from 1 to 61. 
Count by 3’s from 2 to 62. 
To be read by the teacher. 
Write in Arabic: 
Peey7050.. 2,010,002. 92. 40,200. 
425,209. 5s (10,010. ~ 6.7 300,040. 
7- 9,610, 684-3 1.6; g. 215,000. 
10. 80,090. II. 600,000. 12. 809,740. 
13. 100,010.14. 916,008. 15. 835,941. 
£02970,000." 14. 90,005. / 18. 5,016. 
19. 213,033. 20. 30,000. 
Write these or similar numbers on the black- 
loard and require pupils to read them. 
For rapid solving. 
64+4+3+2+4+4—-3=+4x6—3—3+4 
+4? Ans, 32. 
Beet tes 1 3 1423 > 5 XA 4 1+ 313 
4+ 3t5 +5? Ans. 3r. 
Coe te dt 4s +34 tat 25 +2 
+3+3+4? Ans. 53. 
Ae es 8a ec 23-34-4342 
+3+3+22% Ans. 44. 
eg io 21 44 4 + 3-2 +334 +4 
feeat satis 4c eh ANS. 52. 


84 


FIRST STEPS AMONG FIGURES. 


34+24+44+34+54+3+2+4+4+1+244 


+14+3+1+2? Ans. 44. 
44+-343744263 +263 +3 tegen ee 
toi 2 Domai 
34+24+41-4434+3+2+24 35 
+3+2+1? Ans. qt. 
1794+4+3761615—3--3 460 
3-5%X3! Ans. 12, 
9+44+3+2+3 X5—4—3—2+4—1+6 
X3+3+2 2. Ans, 17: 
16+4+31+3+3+2—1+5+3+3-3% 
5—-3+4—17%4? Ans. 5. 


In giving the following examples as well as 
those “ for rapid solving” the teacher should be 
very careful that pupils do not acquire a 
pernicious habit of counting instead of adding 
at sight or as soon as heard. 


Slate examples. 


(t.) (2) (3) (4) (5) (6) (7) (8) (9) 


ml wwe Heenan 


I 


ta | 
Yel ay Cy eerep 
~~ 
N m= WS NO = BO = = NY 
= 
“SAWN edcyeps SUC ee TO ee eee 
ert PG Tea eS 
Wooo iN ieee 
Ca artnet 
i 
TN Gy eae 
— 
BS, | == N NY = NN NN A 


a 
pa 
i 
an 
Le | 
oul 


85 


(10) (11) (12) (13) (14) (15) (16) (17) (18)(19) 


FIRST STEPS AMONG FIGURES. 


(24) = (25) 


(23) 


) 


(22 


(21) 


(20) 


20 


27 


? 


*9 


29 


#3 


86 FIRST STEPS AMONG FIGURES. 


(26): (27) (28): ; (29) (30) (gee 


Z I 2 I 2 2 2 
I 2 I 2 I I ce | 
2 I 2 I 2 2 2 
2 1 2 2 | 2 2 I 
2 2 I I 2 2 2 
2 I 2 2 I I 2 
t I I I 2 2 2 
2 I 2 I I I 2 
I 2 I I Z I - 2 
2 2 2 2 2 2 I 
2 I 2 I I 2 2 
2 I 2 2 2 I I 
2 2 I I 2 v2 2 
I 2 2 2 2 2 2 
—_— I I I a 2 2 
a4 my = 7 24 ee. its 


21 24 21 25 26 


FIRST STEPS AMONG FIGURES. 87 


(33) (34) (35) (36) (37) (38) (39) (49) . 


I 2 I 2 I 2 I 2 
2 I 2 I 2 I 2 I 
2 2 I 2 I 2 I I 
2 I I I 2 2 I I 
I I 2 I 2 2 I 2 
2 I I 2 2 I 2 2 
I 2 2 2 I 2 Zee tt 
I 2 2 I 2 2 Ins) 2 
2 2 2 2 2 2 2 2 
2 2 I 2 2 2 2 2 
2 I I I I Det I 
2 I I I 2 2 I 2 
I 2 2 I 2 2 te | 
2 I I 2 I I 2 2 
I 2 2 2 2 2 2 I 
2 I I — 2 7 I — 

marae. Se 23 a. Tae Bie tik Oo 

ZO TA «23 27.5227. 25 


If the teacher is careful that the pupils do 
not keep any of the solutions of the foregoing 
examples, they may be given 2 or 3 times 
over, first solving them all, then solving them 
aJl again and so on 


8% FIRST STEPS AMONG FIGURES. 


(41) (42) (43) (44) (45) (46) (47) (48) (49) 


2 23 #39 413 632.) 643) Sao 
3 32 23 .2T <13 033° $20 
1 632 «6 3E 33" 327. 30" 833 
Q° 2k 12. 10. “21% 12.0023 
1.39 23 32 6°33 «4.33 SS Ea 
3.12. 32) 13° $20, 225) 3 eae 
% 63% -33. “225 32 431 oc ee e 
2°23 2b 31). 23 5513S cee 
3. 33 29 23° 33 3305933) eee 
20 217 225 198 239 230 233 2205205 


(so) (5t) (52) (53) (54) (55) (56) (57) 
32.21 31. 23 © 30° 23 agmmeeme 
Phe Kee oe 31.. 23° 3) 2i0eeee 
33. 20 32 33 «610 | -23)segaieee 
220 ts eso 30. 22-530 Rai gues 
12332) 13 22 31, 220). 20a 
3h. 33. 31 - 93. 33> SG 
33 a1 235. 312 23 (jase same 
23 13 32 23 I2 21 vik 23 
30°. 32, 33. 305133.) (33ceuntG mune 
23 31 23 23 22 23 32 33 
13°33: 33. (33. 320. 320) Samu 

274 282 274°° 202 “271 (283°. 260qmeun 

See P. Ed., p. 33. 


a eee -_—— ees —_—— eres 


FIRST STEPS AMONG FIGURES, 


89 


(58) (59) (60) (61) (62) )63) (64) (65) (66) 


21 


33 
20 
13 
2 
23 
32 
21 
3 
32 
23 
31 
12 


= 


306 


33 
23 
31 
20 
32 
13 
oo 
FANE 
32 
12 


33 
21 


23 


—— 


327 


23 
31 
13 
32 
20 
33 
21 
13 
32 
Ze 
30 
13 
32 


315 


30 

23 
33 
20 
13 
32 
23 
I2 
ZZ 


Ae) 
30 
21 
33 


326 


33 
21 
13 
30 
23 
32 
II 
33 
23 
32 


13 
22 
31 
33 
21 


I3 


ar 


33 
20 
13 
oy 
23 
31 


32 
13 
31 
23 
33 
ZI 
13 
To 
32 
31 
23 
32 
31 


21 


33 
32 
23 
30 
22 


33 


12 
32 
23 
31 
13 


3 


33 
22 
31 
23 
31 
12 
23 
32 
31 
23 
r3 
33 
ee, 


317 325 328 339 


Teach pupils to prove every example in ad- 
dition by adding both upward and downward, 
and in this way they will get more practice— 
just what is needed. 
_ If the pupils have their books—P. Ed. —the 
following examples are intended to be given at 
recitation for immediate solution, while those 
in the P. Ed. may be solved by the pupils at 
their seats and brought to recitation, 


go FIRST STEPS AMONG FIGURES. 


ee 


If the pupils have not got their books these 
examples may be written on the board or read 
to pupils to solve at their seats, a few daily: 


(TSE) (3) (4) (5) (6) 
231 321 333 331 132 133 
322 233 223 223 223 332 
122 213 Bie 113 330 i. sign 
333 332 233 322 233 322 
212 322 213 232 213 331 
223 ae. e198 223 131 332 213 
232 322 132 303 213 329 
221 222 a2 233 233 a 


2,096 2,109 2,091 1,888 1,910 1,808 


(7). (8) ©) (9) (10) 2911 |e 
22 eye 213 123 2a5 22 
203 re a oo. 37s 
332 ZOE aat 222 232 124 
P2SCr Bae 233 233 L2n 232 
333 333 SOF eo 20a 
212 232 23 312 323 233 
See cea 234 233 ee aay 
Toke! 250 ee oe DES tse 
223 130 2d5 4e5 Las es: 
331 412 321 312 ai 212 


2,445 ~2,663) 2,022 2)543°> 2552 Sue 
Review these if need be; in any case be sure 
the pupils can add such examples as the above 
readily and accurately. 
See P, Ed., p. 36, 


FIRST STEPS AMONG FIGURES. 


Examples in subtraction. 


13. 69758 
6035 


63,723 


17. 75860 
2330 


733530 


14. 97856 
74239 


23,626 


18. 79685 
4242 


75443 


Multiplication. 


21. 32032 
2 


64,064 


25. 23103 
2 


46,206 


22.123103 


3 


69,309 


26. 14023 
2 


28,046 


15. 69587 


20152 


495435 


19. 58796 


25062 


331734 


23. 24130 
2 


48,260 


27. 32023 
3 


96,069 


29. 40312 
2 


80,624 


30. 31203 
3 


93,609 


ob 


16. 68059 
6040 


62,019 


26. 96807 
4200 


92,607 


24. 31402 
2 


62,804 


28. 20312 


3 


60,936 


92 
Division. 


31. 2)48206 


24,103 


34. 2)28460 


14,230 


37. 2)84602 


42,301 


35. 


38. 


FIRST STEPS AMONG FIGURES. 


- 3)90396 


mn 30,tt2 


3)39069 


13,023 


3)30960 


10,320 


(1) 
324 
431 
243 
432 
444 
342 
g32 


2,448 


(2) 
434 
431 
444 
213 
332 
441 
424 


25759 


(3) 

23 
431 
344 
244 
432 
421 
444 


25339 


See P. Ed., p. 38. 


(4) 

43 
324 
442 
243 
134 
424 
233 


1,843 


ee 


33- 3)69306 
23,102 

36. 2)60482 
30,241 

39. 3)93600 
31,200 

(5) (6) 
342 5-0 4at 
434 343 
4457) 242 
321. 434 
443.,...gnhd 
22 Ael 
314550 Gee 
1,900 2,429 


FIRST STEPS AMONG FIGURES. 


(3) 
31 
42 
44 

4 
31 
43 


(9) 
422 
344 


(10) (1x) ‘(12) 
ich hie Shoe Y 
323 44 44 


See el See oe See cae 


2,573 


2,953 232 2,834 


94. FIRST STEPS AMONG FIGURES. 
(28)~(29) (30) (31) (32) iggy 
243. «21 «44. 423° 343 gee 
444. 432 21 341 . 444 5025 eeeae 
231. 6 344). 34,444 fea 4 44 
422.211. 43, 332 4432 (eeaqeeeete 
444 321 AZ 2 t feel 22 22 
233 432.31 441 343 eee 
423. 343° 24: 3824. 444 S039 
344. 444 32 432 (434° 9 140 gee 
212 323° 43 344 | 232 5a geen 
440. 342 44.9 443 9 347 4 eee 

3,436 3,213 359 3,737 3,166 291 347 
(35) (36) (37) (38) (39) (40) 
343. 423° «14 «314° 232, Br ge 
444 442 3 432 443 “429443 
321 341 40 341 444 43 344 
432° 234 34.¢ 134 $9324 eee 
344 434 . 42° 244 | 437 ages 
344 6340 > 447-440 3435 4 
234.423 +. 34, 342. 234 gee 
421. 412 . 31 444 444 Sea 
432 344 > 23. 233 . 323 42a 
444° 431. 42) 132450242 eae 

3,759 3,825 308 3,249 3,460 317 2,911 


See P. Ed., p. 42. 


FIRST STEPS AMONG FIGURES. 95 


(42) (43) (44) (45) (46) (47) (48) 
Beemesa ear 24... 234 Pe EY, 
Perm aes 4s dar yy, dg 
231 BimAd ae gage? Gy 422 
eee 4 a4 324. 24) < 334 
344 1:42 123 Heett > BLA 244 
Seen 6.442 )) 33-6 442.6 44 6413 
eran ad 3440 24 7134 2 43-423 


Brioemmsz2 301° 394 3:853 ~ 373 3,952 


b 


a Cc d e 
4735|2473|/2647/3526/4735 
345617456134561734510734 

f g h 
2647)352/665 
gH 7314551737 

For subtraction. 

a b c d 
119812]105914|10697]|1211 107 
O74 51 43° 7} 6345| 5 7°34 

e f g 

10711 8]129138]9810611 13 

Reagereo G7 OA Si3a7 4 5 6 


96 FIRST STEPS AMONG FIGURES. 


For division. 


a b Cc 
I2 25 12 21 | 28 36 49 24 | 9 20 10 42 
65 4.31 7 © 7.6) 354esee 
d = f 
8.35 15:8 | 30 21 42 6 | 2420 16 35 
3.734155 7. 63:54 s5Reae 
oa h 
16 14 12 | 28 15 30 
4.07 34 4 5e5an8 
(1). (2) 343) Se eee 
342 24. 2432 404 23 
421 32 434 34 34 
34 44 on 2a s ead Be 44 
43° 3 343 342 3 


143 42 Se ae 43 
432 33-2234 aaa 4 
341 4. 444 342 44 
ee 13 321 244 32 


4,604 452 4,296 4,774 476 
See P. Ed., P- 40, 


FIRST STEPS AMONG FIGURES. 97 


6. Add 342, 234, 344, 421, 342, 24, 431, 
231, 4, 423, 344, 434, 324. 

7- Add 24, 43, 24, 32, 44, 34, 42,4, 34, 
21, 43, 14, 34, 23, 43. 

8. Add 43, 233, 424, 341, 3, 434, 4, 342, 
434, 342, 243, 414. 

g. Add 2, 34, 44, 41, 32, 43, 34, 3) 44, 34, 
23, 42, 34, 41. 


How many quarts are there in three gallons? 

Solution: In ove gallon there are 4 quarts, 
in three gallons there are three times 4 quarts 
or 12 quarts. 

The following examples may be given during 
recitation : 

1. How many feet have 5 horses ? 

2. Arthur was paid 5 cents for doing an 
errand and his sister gave him q cents ; how 
many had he then? 

3. If a carpenter can drive 3 nails in a 
minute, how many minutes will it take him to 
drive 18 nails ? 

4. Charles had 10 snow balls in a pile ; he 
threw 4 of them at his playmates. How many 
remained in the pile? 


7 


98 FIRST STEPS AMONG FIGURES. 


5. Mr. Smith paid 4 dollars for the cloth 
for a pair of pants and 2 dollars for making 
them ; what did the pants cost him? 

6. How many vests at $4 each can be 
bought for $24? 

7. At 4 shillings a pair, what cost 5 pairs 
of scissors ? 

8. If a hat cost $3 and a pair of boots $10, 
how much more do the boots cost than the hat ? 

g. Andrew has a pair of ponies, how many 
feet have they ? 

1o. How many more feet than eyes has a 
four-horse team ? 

tr. How many less heads than feet hasa _ 
three-horse team ? 

12. A boy spent 21 cents for marbles at 3 
cents each; how many marbles did he get? 

13. Barton has 17 cents ; how many pencils 
at 2 cents each can he buy and keep 3 cents? 

gt5+44+5+5+3t4t5+3+4+2+3 
+2+4—% “Ans. 52. 
5St5 734324141543 4244 ee 
+4+5+3+4=? Ans. 60. 
Examples like the above, having only addi 
tion, may be given both forward and back- 
ward, thus they will make 4 examples instead 
See P. Ed., p. 45. 


HI lo 


FIRST STEPS AMONG FIGURES. 99 
of 2. Still more may be made by commenc- 
ing with another number, as 12, and adding 12 
to the answer. Thus: 

12+ (5+5+313+4444+54+34+2+4+ 
I+ 5+4+54+3+4)=60+12=72. 
8+4+5+4>3X5+4+445—-3—4—5 

—A—5—3-—32? Ans. 2r. 

I5+44+4—-5>345+5+4+4+7+3+4 

r5+5¢ Ans. 29. 

Seems 4 ter t4atsot3ters 5 

+4+4+3? Ans. 58. 

See StS tits t4teretsgto+4 
c+ 4+4ts? “Ans. 62. 
$4+51+44+512+4+4%X3—5+4X8—4 

—5—5—4? Ans. 14. 

Re O47 4-77 X65 3-5 — 

4+2+5=6+5? Ans. 9. 

I8+5+4+4+5—-3—-4—-5+4+5+5+ 

eee ADS, 27. 

gee 5 344 42 5 + 34 
+5+3+3+2+4? Ans. 61. 

Seo 4 3-4 X75 —2—-3—4—-5 3 
—5—4X3+3+6? Ans. 4. 

4424+5434345+444+5+5+3+2 
eae tet 4-1-5 3 © Ans, 07. 

Bt4+44+5t+24+543434+4+5 +444 
fe Sates tose sa ApS. 68, 


100 FIRST STEPS AMONG FIGURES. 


34415 437475751474 4555 


+5+3+5+44+5? 


Ans. 69. 


5445444541176 +375 
Ans. 12. 
744457514474 13--5 34S eee 


54 Reon ae 


—4x3? Ans. 21. 
§+34+5+47473 13757533 eee 
—5—5-4—4—-3? Ans. 6. 

(x) (2)... (3) (4): © {SC 
35 5 04 4 59 5a eee 
54. 43.-°35 235 (S45 © 53aeae 

5.54. 43 4: \45> . 24°50NSs 
43°. 935 5.53. .335. 45a 
24... 42. 52 45> 4 Slee 
555i 4g” 24 244 Gee 
34° 35 9525 55 > 32)” 54a 
§ 22> 64 G5 Soe 5855 
45 45 21) 44.340 eee 
347 364 332 341 385 344 361 

Examples in subtraction: 

1. 67,548 2. 69,584 3. 75,897 4. 97,867 
43:235 345331 545353 52,343 
24,313 34,253 21,544 45,524 

5. 64,786 6. 79,684 
2,432 72953 
62,354 72,631 


See P. Ed., p. 47. 


FIRST STEPS AMONG FIGURES. rot 


The teacher may use his own judgmeni as to 
teaching subtraction when some figures uf the 
subtrahend are greater than the corresponding 
figures of the minuend. — 

A purely mechanical method is here given 
with the idea that the method of doing many 
things may properly precede the veason for the 
method. 

If the following method be used, after one 
or two years or in a larger book the reason of 
the method should be fully explained to the 
pupil, and he should then be required to give 
the reasoning himself. 

32,413 — 5,667. Solve by separating the 
figures of the minuend, as in the line below, 
and then when any figure of the subtrahend is 
smaller that the figure above it, write 1 before 

era T4153 1.3 
it thus : 5 6 6 ¥ and then subtract, 
2 Se ly pea oa ae 
being careful when 1 is prefixed to the upper 
figure to add 1 to the next left hand figure of 
the subtrahend. The following examples are 
so arranged that no figure of the subtrahend 
is greater than 7, the tables having been 
learned only so far. After solving the above 
example the pupil should say 5,667 from 32,413 
leaves 26,746. 


102 FIRST STEPS AMONG FIGURES. 


ee 


7. Qto201 8. 423423 g. 621423 
56454 45466 53667 
853747 377957 567756 


As soon as the pupil understands the mechan- 
ical work, he should not be allowed to write the 
1’s in the minuend, but imagine them to be 


there. 


10, 831,242 “II. 513,423) 2912. 430pees 


63,567 46,647 262,637 
767,675 466,776 167,576 
13. 731,420 14. 342,031 15. 532,514 
54,654 25,266 65,251 
676,766 _ 316,765 467,263 
16. 731,420 17. 624,091 18. 831,042 
61,265 62,035 63,415 
670,155 562,056 767,627 


See P. Ed., p. 48. 


FIRST STEPS AMONG FIGURES. 103 


(2) (2) are t) ce es) (9) 
eyes 32. 523 249 823555544 
Boe 455. 454 Sad 3251 S52 
pom 44. 342-44.) 534.) 435 
ee 503-535 Bonaeere 55 2534 
meee 434. 453 Ripe ots. 6253 
Ree ss5i 544 fs - 45. 424) 6 545 
43 Been 325 5 4ue ak hang 324 

See 545. 454 Bh ESA Ge: 
poess4) 543 Agee 444e vys43 


335 3,205 4:173 = 384 3,876 3,962 

More examples may be made from these by 
reading them from the center each way,thus 
giving new combinations, or by giving two ad- 
ditional numbers, one above the upper number 
and one below the lower one; in this way the 
combinations will be different whether the 
pupil add upward or downward. 

Of course the teacher must add the sum of 
these two numbers to the answer in the book to 
get the answers of the new example. Exam- 
ples in Pupils’ Edition may be treated in the 
same way. 


7. 30,142 8. 23,103 9. 42,301 10. 24,130 
2 3 2 2 


— oy —= —> — ——_ —_— 


60,284 69,309 84,602 48,260 


104 


FIRST STEPS AMONG FIGURES. 


Show the pupils that in multiplying, as in 
adding, if any result is greater than 9, the left 
hand figure is added to the next result, which 
is of the same kind. 


II. 63,524 12. 36,546 13. 36,426 14. 63,524 
3 2 3 4 
190,572 73,092 109,278 254,096 
15. 26,463 16. 64,524 17. 53,625 
4 4 5 
105,852 258,096 268,125 
18. 26,463 19. 46,035 20. 26,304 
5 4 6 
132,315 184,140 157,924 
21. 25,036 22. 50,264 
4 6 
100,144 301,584 


Caution: Do not allow the pupil to write any- 
where what he is to add to the next product. 
See P, Ed., p. 52, 


FIRST STEPS AMONG FIGURES. 105 


For addition and multiplication. (8 and rev.) 
d f g 
84736 36258|47362 
67834 45678|34567 
i 
362 
678 


Cc 
73625 
78345 

h 
5847 
8345 


(e 
25847 
56783 


a 4 
58473/62584 
34567 \83456 


For subtraction. 


a b | c 
Berspicowt4 5 Sera 
eee 3) 4503 0 454) 3° 8 7 6 6 

d e f 
rs Is PaeOnTO | Feb 1e.4T2 ieee "3 
(ems 415.6 7 8 16 54. 3 

g h i 
o-oo 
Beas. 0 719.3145) 67 8 

For division. 

a b é 
24 49 24 18 8|40 20 6 48 21/42 20 32 15 56 
Seems 4|-5 + 43°98. 71.65 4 3 8 

Tees e f 
H24,10 30 56/32 28:12 64 35/1230 12 21 14 
geese 7} >. 43°98 7[' 6 5 4°37 

g h 1 
eGmmeeteet0124\40-15 42/15 16-48 28 
Pepe A 3/96 7),5 8 6 7 


106 


(24) 


345 
234 
532 
424 
352 
443 
345 
532 
314 


31521 


(30) 


545 
434 
55 
lee: 
224 
532 
454 
534 
325 
43 


3,199 


FIRST STEPS AMONG FIGURES. 


(25) (26) 
235 25 
543. 54 
455 43 
345 55 
A32n42 
Soqueed 
345 54 
553 65 
345 54 
3,807 383 
(31) (32) 
435 45 
DA. 
345 23 
534 54 
242 2045 
555 33 
3430 654 
543 45 
254 54 
442 33 
4,247 390 


See P. Ed., p. 59. 


(27) 


435 
543 
a4 
455 
543 
235 
352 
544 
535 


3,876 


(33) 


543 
355 
ot 
542 
345 
554 
435 
543 
343 
a0 


4,449 


(28) 


345 
523 
454 
325 
543 
454 
535 
234 
555 


3,968 


(34) 
231 
545 
354 
434 
543 
355 
332 
435 
543 
345 


4,117 


FIRST STEPS AMONG FIGURES. 107 


The following tables are an excellent prepa- 
ration for short division. 

Before the pupils solve the examples on page 
120, give them a review of this page. 


Division with remainders. 


*o5’s (and review. ) 


a b c d e 
SGntO si1\22 9 .6..13/5 14.4, 7118 13. Tz 
3.42 31424 3l2. 43214 3 2 
6’s (and review.) 

a b Cc d 
fagioos 20)11 26.11 Oj21°10 19 7\33 149 5 
Weel 5 4° 3 O15 4° 3 21-5 6 2 3 

e f g h i 
Pimento 71% 5 35 27/20 23 Siig 7 32 
pemeee set 3 2) 9.67,95| 6) 23) 4°56 


1. What cost 8 dozen buttons at 7 cents a 
dozen ? 

2. What cost a pair of boots at $7 anda 
hat at $5 ? 

3. How much more does a reader cost at 6 
shillings than a speller at 2 shilliags ? 


*These are to be recited as follows: 3 in 16, 5 times and 
F remainder, &, 


108 FIRST STEPS AMONG FIGURES. 


4. There were 9g birds in a flock, and a 
hunter killed all but 4; how many did he kill? 

5. How many knives at 7 shillings each 
may be bought for 35 shillings ? 

6. A boy spent 21 cents for marbles at 3 
cents each; how many marbles did he buy? 

7. There are 8 pigs in one pen and 5 in 
another ; how many in both pens? 

8. A boy earned 8 cents on Monday, 7 
cents on Tuesday, and 6 cents on Wednesday ; 
how many cents did he earn in the 3 days? 

9. Henry bought 8 marbles at 4 cents 
each ; what did they cost him? 

Io. 13 boys were skating ona pond ; during 
the afternoon 8 of them fell upon the ice. How 
many of them did not fall? 

11. Fred had 15 cents, he spent 5 cents for 
oranges and 1 cent for candy; how many 
cents had he left ? 

12. George had a bank into which he put 7 
cents, his father 8, and his sister 4; how 
many cents had he in his bank ? 

13. Charles has 6 cents and his sister has 2 
cents more than he; how many cents have 
both ? 

14. Henry had $8 for Christmas and his 
sister half as many ; how many had both ? 

pee P. Ed., p. 56. 


' FIRST STEPS AMONG FIGURES. 109 


15. William bought an orange for 4 cents, a 
fig for 1 cent and some candy for 2 cents. He 
sold them all for 12 cents. How much did he 
gain? 

16. A boy received 6 cents a bushel for 
picking hops ; he earned in this way 48 cents 
in one day. How many bushels did he pick? 

17. Carrie whispered 3 times in 1 day, for 
each time she whispered she had to remain 
after school 5 minutes ; how long did she have 
to remain ? 

18. 20 cents are to be divided equally 
among 5 boys; how many cents should each 
boy receive ? 

19. If 6 pieces of tape cost 24 cents, how 
much did one piece cost ? 

20. If a boy earned 28 shillings in-7 days, 
how much did he earn in 1 day? 

21. Samuel walked 28 miles in 4 days; at 
that rate how far would he walk in 1 day? 

22. How many quarts of milk at 6 cents a 
quart can be bought for 36 cents ? 

23. If 4 gallons of molasses cost 28 shil- 
lings, what cost 1 gallon ? 

24. 56 cents will hire how many boys for an 
hour, if each boy is to have 7 cents for an 
hour’s work ? 


110 


(1) 
52 
45 
3! 
53 
44 
25 
53 
44 
a5 
34 
55 


od 


471 4,640 4,481 


(11) 
544 
354 
435 
543 
431 
303 
454 


FIRST STEPS AMONG FIGURES. 


(2) 
242 
525 
343 
451 
535 
434 
453 
345 
523 
2 5+ 
535 


(3) 

543 
244 
554 
343 
234 
455 
542 
435 
ore 
344 
535 


(4) (5) 


42 
54 
35 
43 
51 
45 
34 
34 
25 
45 
she 


34 


(6) (7) 
535 344 
454 553 
342 435 
535 234 
453 544 
341 345 
355 453 
434 423 
544 345 
355 534 
433 453 


(15) (16) 
345 432 


53 544 


434 434 
545 542 
354 344 
534 453 
435, G42 
543 Sao 


See P. Ed., p. 59. 


5,212 5,200 5,086 4,729 5,300 


FIRST STEPS AMONG FIGURES. Il!t 


Pupils read : 


I. 3040321 2. 30245000 3. 463317030 
4. 500030261 5. 2000032 6. 15000000 
7. 320030000 8. 674346537 9. 42000 

10. 400320219 II. 3605000 I2. 50000018 


13. 463308260 14. 75000341 15. IOIOOIO 


Read the following: 


» 67345768 2. 476347854 3. 74000037 
4. 735400005 5. 900007000 6. 316000140 
7. 80370000 8. 700000004 9g. 7020500 
10. 86000045 11. 800006000 12. 90000007 
13. 8060700 14. 90430000 15. 735468371 


Teach pupils to write Arabic to billions, that 
is including 999,999,999. Teach the pupils to 
numerate by periods to the right as well as to 
the left. Thus: units, thousands, millions; 
millions, thousands, units, until they are per- 
fectly familiar with it. 

Method: Suppose the number ten million 
ninety thousand three is to be written. Instruct 
the pupil to write the number of millions first 
with a comma after it, and that the first period 
at the left does not need to be filled to three 
places by prefixing ciphers. For the above 


t12 FIRST STEPS AMONG FIGURES. 


number the pupil will write ro, at first. Teach- 
er ask 10 what? Pupil, 10 million. Teacher: 
What period is next to right of millions? Pu- 
pil: thousands. ‘Teacher: How many thou- 
sands are there (in this number)? Pupil: 
ninety. Teacher: Write it after the comma, 
and, as it fills but two places, place a cipher at 
the left of the 90 and a comma after the go. 
The number will now be 10,090. - Teacher : 
You have now millions and thousands; what 
period is next? Pupil: units. Teacher: How 
many units are there in this number? Pupil: 
three. Teacher: Place the 3 to the right and 
prefix two ciphers to it to fill the three places 
of the periods. Place a period at the right 
because it is the end of the number, and you 
have 10, 090, 003. 

Teach the pupil when writing numbers at the 
blackboard to turn directly away from it as 
soon as units and the period are written, for he 
should be sure that the number is correct with- 
out numerating to the left. 

Teach the pupils of course when there are 
no thousands, to write three ciphers and treat 
unit’s period in the same way. 

When teaching to write billions, trillions, 
etc., follow the same method. 


FIRST STEPS AMONG FIGURES. 113 


Teach the pupils that for the word hundred, 
you will write on the board hun. ; for thousand, 
th. ; for million, mil. ; and when you get so 
far, for billion, bil. ; for trillion, tr. ; for quad- 
rillion, quad., etc. This method will save the 
teacher much labor and much space on the 
blackboard. Thus the teacher may write upon 
the board: 

** Write in Arabic five mil. forty th. six.” 

To be written on the blackboard for pupils 
to bring to recitation written in Arabic: 

1. Write in Arabic four th. fifteen. 

fs) a pe ** twenty mil. three hun. 

ae f) nine milforty th. 


Caution: Teach pupils to put a comma only 
after each period, except the last. ‘Thus in 


Nore.—The following diagram may assist pupils in writing 
numbers, but after being used a few weeks the pupils should 


write numbers without using it: 


| Millions. Thousands, Units. 


35 | 058 


The teacher may draw a diagram like the above and allow 


003 


the pupils to write numbers in it, as the number 35,058 003 
is placed there. Teach the pupils that the third period 
represents millione, and that each period is read as if it stood 


alone, only that its name is given, 


8 


1i4 FIRST STEPS AMONG FIGURES. 


three hundred seventy-five million, four hun- 
dred eight thousand seven hundred forty, the 
pupil may have an idea that he should put a 
comma for hundreds, whereas the above num- 
ber should be written 374,408,740. When no © 
name is given to a number it is supposed to be 
units; ¢. g., two thousand three hundred eight. 
Eight here means eight units, and three hun- 
dred eight, (which has no name given to it) is 
308 in units period. 

The following examples may be written upon 
the board and the pupils required to bring the 
answers to class: 

1. Write in Arabic, thirty million, eight 
thousand, three hundred fifty-one. 

2. Write in Arabic, two hundred fifty thou- 
sand. 

3. Write in Arabic, one hundred sixteen 
million two hundred twenty. 

4. Write in Arabic, three hundred million, 
sixty thousand, five hundred, seven. 

5. Write in Arabic, five hundred thousand. 

6. Write in Arabic, one million, one thou- 
sand, one. 

7. Write in Arabic, seventy million, six 
hundred thousand, eighty. 

See P. Ed., p. 61, 


FIRST STEPS AMONG FIGURES. 4.5 


8. Write in Arabic, one hundred fifty-four 
million, two hundred sixty-one thousand, five 
hundred forty-eight. 

g. Write in Arabic, eight hundred million. 
to. Write in Arabic, three million, three. 
11. Write in Arabic, ten thousand, ten. 

12. Write in words, 809271300. 

13. Write in Arabic, five hundred, four 
million, forty. 

14. Write in Arabic, ten million, ten thou- 
sand. 

15. Write in Arabic, one hundred one 
million, one hundred one. 

The Roman notation uses the following 
Pe eieeeee) ——1,.4V 52 X— 102 Ls, C=— 
100, D=500, M=t1o00. 

To read a number expressed in the Roman 
notation : 

*Rule: Add the values of the letters, observ- 
ing that when a letter is followed by one of 
greater value than itself, the difference between 
the two is to be taken in making up the sum. 

16. Write in Roman, three hundred forty- 
five. 

17. Write in Roman, one hundred seventy- 
four. 


* From Olney’s Elements of Arithmetic, 


116 


18 
19 


FIRST STEPS AMONG FIGURES, 


. Write in Roman, four hundred sixty-two. 
. Write in Roman, six hundred ninety-six. 


nine, 


21 
22 


23 


20. Write in Roman, eight hundred ninety- 


. Write in words, 245306341. 

. Write in words, 32743642. 

. Write in Roman, three hundred eighty- 

seven. 
For rapid solving. 


I. 


44+34+54+2+64+44+54+64+34+5+4 
+3+6+6+5=? Ans. 67. 


» 34+54+446434+6454+24+64+5+4+5 


+4+6+6-2-b4— i Ansa 


~5+64+64+44+34544+3+6+246 


+6-+5 +64+-2--4= fA sae 


© 4434675 +24+-64-64-'5 + Anes ae 


+5+4+6+6+5=? Ans. 76. 


-54+64+344+64+5+64+64+5+4+3 


+6+6+6+5+4=? Ans. 80, 


© 16 +657 X54+35-3XS8 35a 7 oe 


+5+3—5—6=?. Ans. 29. 


- 19+4+5+4x6—6—5—6+5 X8—4 


+6+5—6x7—4=17 Ans. 31. 


. 7X6+5+6—4—6—6—5—SxO75 


+6—3—6—5+6=? Ans. 27. 


. 6x8+ 6—2—5—6—5—4+4X7—5 


43 OU aaa, Ans, 8, 


10. 


II. 


BY 


13. 


er 4s 


re 


16. 


17. 


18. 


19. 


20. 


FIRST STEPS AMONG FIGURES. EY7 


84+77+6+5+5 +475 x6—5—6—6 
—6—3+8+5=? Ans. 7. 
e505 — 6-4-5 +045 +4+6X5 
—6—5—6+5=? Ans. 23. 
47+5—6—4—5—2+5 x 8—5—6—6 
+3+6+8+6=? Ans. 21. 

2a2-5 0-4. * 0+5--0-5=-7.%5—6 
—6—-6x8—6=7 Ans. 18. 

16-4 x6+5—4—6—5+6+51+6+4 
7X S—4—A—5 = ¢  AnSy 27. 
Gee 0 — 4-3-3 040-4 
5-6-5 —6— 2-7-5 =? 2Ans. 3: 
Boe O 4-0 Be 4 a-O 5 0-4-4 X'S 
—=6—5-+-38x7= 7? Ans, 21. 
61—6—6—5—5—4—-6—3-—6+5 x8 
—5—6+7+6=? Ans. 9g. 

Bate O55 tt 3 6 54 Ot 37717 
+5+5+6+6=? Ans. 33. 
16+5+4+3+5—6—3+8x5—6+4 
Han 5-6 —4— +) yAnS. 13. 
27+5+4~6x8—6—4—5—3—6—5 
—4+3xX6—4—4—6=? Ans.16. 


The teacher is advised to give a few exam- 
ples in subtraction each day, and with them a 
few in multiplication and perhaps in division 


also, 


118 


I. 


FIRST STEPS AMONG FIGURES. 


Subtraction. 


3,423,056 
654,368 


2,768,688 


- 4,320,032 


543,054 


3776,978 


- 7,360,042 


2,500,075 


4-859,967 


10. 8,340,050 


762,034 


_7;578,016 


13. 6,343,520—656245= ? 


2 i fe 


834,657 


6,517,386 


. 63,140,052 


6,572,036 
56,568,016 


. 61,420,035 


549,257 


60,879,778 


64,230,051 
510,765 


63,719,286 


» 75352,043 3. 9,635,024 


eae 548 


8 8,888,476 


- 8,400,314 


50,248 


8,350,066 


- 5,304,036 


202,356 


5,101,680 


12. 75,310,040 


230,076 


75,979,964 


Ans. 5,687,275. * 


14. 94530 062—8,240,277=? Ans. 86,289,785. 
See P. Ed., p. 64. 


FIRST STEPS AMONG FIGURES. 119 
* Multiplication. 
1. 648,057 2. 746,805 3. 470,685 © 


8 7 6 
5,184,456 eee 2,824,110 
4. 8,640.753 5. 680,574 6. 358,407 
_ 7 8 6 
60,485,271 5.444.592 2,150,442 


7m055,740 X4— 2 Ans. 2,742,960. 
8. 7,406,853 x7=7 Ans. 51,847971. 


The teacher should solve an example in 
which there are two figures in the multiplier. 


G.0)364.%23= 2 Ans,.146,142. 

Boast 402 % 32> ¢- Ansi-3,7.10;754. 
ot. 20 A25°% 34= ? Ans. 1,238,450. 
Berets 2.025 x 43=— 4 ¢ Ans. 15,205,875. 
Peese.524 x O5—' -Ans..36.629,060. 
14. 350,264xX36=7% Ans. 12,609.504. 
Beer so. 204 x 34 —- Ans, 17,004,336. 
16. 640,536x64=2 Ans 40.994,304. 
17. 4675 X45=—? Ans. 210,375. 


*The numbers used in these examples are pointed off in 


periods for convenience in copying to blackboard or slate, 


126 FIRST STEPS AMONG FIGURES. 


Before taking up examples in short division 
review division with remainders, p. 107. 

In teaching pupils short division when there 
are remainders during the operation, write the 
figures of the dividend well apart ; thus, in the 


: ste 43.7 19 35 33 11 
example 379531~4, write wera 6S 


write the remainder before the next figure as in 
the example given. After solving 2 or 3 exam- 
ples in this way, write the figures closely, in the 
usual form on the blackboard, and let a pupil 
divide orally, the teacher using the crayon, one 
pupil telling how many times it is contained 
and what remainder, the next stating what the 
next partial dividend is and how many times 
the divisor is contained and what remainder, 
etc. 

Let the pupils first solve the examples with- 
out remainders, given in Pupils’ Edition, and 
the following 4 examples : : 

I. 24129318—-3=? Ans. 8,043,106. 
2. 281,683,220—4=? Ans. 70,420,805. 
3. 12,246,921 ~-3=? Ans. 4,082,307. 
4. 322,412,836+4=?P Ans. 80,603,209. 

With remainders. 

5. 83,923-3=? Ans. 27,9741. 
6. 182,539+4=? Ans. 45,634.34. 
oo) fg) Wear aie 


and 


FIRST STEPS AMONG FIGURES. I2I 


7- 1,373:224>3=% Ans. 457,7418. 
8. 91,354.328+4= ? Ans. 22,838,582. 
Oma73.701-5—!f, Ans. 54,756". 

10. 1,035,879+6=? Ans. 172,646°. 

T1. 2,683,507+4=? Ans. 670,876%, 

12. 3,921,278--6—7? Ans. 653,546%. 

Method of teaching pupils to add numbers 
like 46 and 7. 

In adding 46 and 7, ask the pupil what he 
should add first, and either get from him or 
show him that 6 and 7 are to be added first; 
that it makes 13, of which the right hand figure 
is 3, which will be the right hand figure of 
the sum of 46 and 7, and that 46 and 7 are 
53. Persevere in this plan upon the following 
numbers or until the pupil in adding such num- 
bers as 37 and 8, will say at once “the right 
hand (or least) figure will be 5 ; 37 and 8 are 


45+6? 67+5? 58+7? 65+8? 
86+5? 58+6%2 34+7? 57+52 
26+8? O57 & a7 +82 78+5 2 
86+7? S74 58+8 ? 63+7? 
47+7? 28+4? 76+5? 53+8? 
7 t:0;? 764672 67+3? 384+7? 
apron 3515 20 47-462 28+3? 


¥22 


FIRST STEPS AMONG FIGURES. 


43 +7! 64+4° 45+8? 26+5? 
4720 38+6! 533s 7447 % 
$5ck 42 he yOnrRoy 37+5?2 58+2? 
5376? 24+3? 65 -+7< 86+ 4? 
27+8? 68+5?2 83+2? 54+6? 
65+3? 36-472 sy 4a 68+8? 
43 +5? 54+2 2 35+6; 16+3? 
Ag 4 38+4? 53+8? 24+5% 
45+2? 86+6? ~47+3? 58+7? 
4944200 3443209 075-5 eae 
Hy Bw 48+3? 


The exercise above should be most used un- 
til the pupils are perfectly familiar with the 
combinations given, which embrace all between 
8 and 3 inclusive and some are given twice. 
It may first be given to the pupils in the order 
above, then commence in the middle of the ex- 
‘ercise and go each way. It should not be 
written upon the blackboard but recited orally 
from the reading of the examples to the pupils. 

1. A boy bought a top for 18 cents and sold 
it so as to gain 7 cents; what did he sell it 
for ? ‘ 

2. James’s mother gave him 30 cents with 
which to buy oranges. At 6 cents each how 
many could he buy? 


: FIRST STEPS AMONG FIGURES. 123 

3. Willie said he had 4 cents ; John said he 
had 4 times as many; how many had John ? 

4. George had 8 sticks of candy and his sis- 
ter had 7 ; how many did both have? 

5. 18 ripe peaches were on a tree and a bad 
boy stole 7 of them ; how many were left? 

6. A flock of 18 birds lit upon the ground ; 
a hunter shot rz of them; how many were 
left? Ans. 11. 

7. How many fingers have 4 boys? 

8. Charlie’s mother gave him 7 cents, and 
his sister gave him enough to make 13 cents ; 
how many did his sister give him? 

g. John had 10 cents, one of his sisters had 
7 cents and the other had 6; how many cents 
did both the sisters have ? 

10. Mary. bought 2 yards of calico for a 
doll’s dress ; she gave 8 cents a yard; how 
much did the dress cost? 

11. A tired school teacher struck a naughty 
boy five times upon each hand ; how many 
times did she strike him? 

For addition and multiplication. 

g’s (and review.) 


b 
74859 
94567 


d e f | g 

6374 | 85963 |74859|62748/59637 

8945 | 67894)56789|49678|94567 
See P. Ed., p. 74. 


a 
85963 
45678 


Cc 


124 FIRST STEPS AMONG FIGURES. 


h i j 
4859 6374 
8945 6789 
For subtraction. 
a b c 
Il, IO631 13 13 15 ;104t2 It 16 14 12 
8 99 5 7. © 5.4: 500 yee 
d e f 
Llei eagle Octhr7zie IO “LONtiees 
4.138.750 7 5aeO ates 6 8 6 
g h 1 
8 15 13 18 TO clades © 9 13 15 IO 
5 Ps] Bee 4-94" s 69 8 7 
j k 
I2 14 14 12 17°9 
3557 8 94 
For division. 
ea b Cc 
42 54 25 32 24 63 16 40 30 63 48 
ThE ord. S09 5 7 
d e f 
27420472 025 48 20 54 12 35 24 64 
9 4 7 659 4 57 0ra8 
g | h i 
28 42 18 24 81 40 56 45 36 30 
7 9 4 9 B27 9 4 5 


FIRST STEPS AMONG FIGURES. 


7 


(1) 
36 


j 
18 49 32 72 
6 8 


k 
18 36 56 
Se Sa 
palpi oe(3)s2 "05 (4) 


654 245 45 
343 634 56 
565 453 43 
654 546 64 


345 563 36 


4,056 4,797 421 
See P. Ed., p. 77. 


426 FIRST STEPS AMONG FIGURES. 


(12) (13) 


(14) (15) 
45 546 
54 354 
66 66s 
36 - 236 
45 565 
56 653 


(16) 
544 


654 | 


565 
666 
325 
456 


302 2,019 2.210 


(3) (4) 
24 15 
45 344 
54a se 
53 544 
44 355 
35189243 
33 5aei25 
42 544 
Chae at 
33. 554 
2 Leaeaas 
BA eetz3 
35 22345 


507 4,800 5,880 
mee 1.) id... p. i 70s 


5.245 5,048 


(17) (18) 
45 53 
6 64 
55 55 
63 46 
34 = 333 
65 56 
268 307 
(6) (7) 
36 26 
56° 353 
63 66 
25 32 
66 3865 
54 66 
46 26 
35 63 
63 66 
56 2755 
6-43 
63 66 
45 35 
614 662 


FIRST STEPS AMONG FIGURES. 127 


The answers to the examples are at the end 
of the book. They are placed there so that if 
the teacher wishes any pupil of the class to 
copy the examples on the blackboard for him, 
it may be done without the pupil’s knowing 
what the answer is. If the teacher prefers to 
have the answers with the examples he can 
copy them from the end of the book. 

In adding long columns of figures it is well 
to write the sum of each column 56 
separately, as follows, so that in 34 
adding each way for proof the 120 
sum of each column may beseen = 57 
at a glance. 

69396 

Show the pupils that when g is added to any 
number the unit figure of this sum will be one 
less than the unit figure of the number, thus: 


9 +37 is 46, 74+9 is 83, &c. 


34+6? 75+9? 26+5? 4748? 
18+4? 47+9%? 24+3%? 85+6? 
36+9? 67+5? 48+8? 294+4? 
64+72 45+3? 26+6? 87+9? 
68+5? 59+8? 74+4? 8517? 
564+3% 47+6? 78+9?% 69453 


128 FIRST STEPS AMONG FIGURES. 


74482. 35442 56471 G7aeoe 
78+62 5o+9? 75+4? 45+8? 
26+4% 574+7% 78+3?% 59+6? 
84+92 65+5?% 4648? 8744? 
68+7? 494+3? 


Give much practice on the above exercise ; it 
will be of great use to the pupil in all additions. 


8. 463,075 x 465 =? 9. 640,753 X 3061 
10. 560,423 X 204= ? 11. 67,052 X224e= 
12. 574,003 X27=8 13. 867,534 X 56=? 
14. 680,574 X 78=? 15. 475,306 X 406=? 


Before taking the next examples give the 
pupils a thorough drill in division with re- 
mainder. 


Division series with remainder. (6 and rey.) 


a b c 
23 10 3125-5 2Zslled 19 27 2207an7 
4 32 66 5 4 23 45 tae 
d e f 
21 817 20 532i 7a | 6 13 15 26 
65 4 3 2-6 5.2 |g 
g 
1549 39 12 
432 6 5 
See P. Ed., p. 60. 


FIRST STEPS AMONG FIGURES, 129 
7’s (and review) with remainder. 
a b | Cc 
22 19 42 34 69 19 14 39 28 62 17 39 
Bae 5. 6 Hee SM i ye A 
or | e | f 
32 23 53 13 34 29 57 47 | If 31 21 52 
es | 45 67 4 5 
P| ages: | 193 
51 28 27 17 47 32 26 21 49 39 25 
(ly py SU) URS ie ae ad On, a 
8’s (and review) with remainder. 
a | b c 
55 204524 | 3416 31 53 29 43 21 29 
Bee | 404° 3 8) 7 65 4 3 
d e | f 
41 69 13 31 20 39 28 53 | 76 47 22 38 
Leow se + Sy ie Siete & O85 
g | h | i 
26 28 63 32 49 33 61 39 | 58 31 14 23 
aeoeey | 25 6-7 8 6 4 4 3 
j k 
19 39 69 | 44 2617 
On) 8053154 


I. 1,396,897 -3=? 
3- 2,144,698 4=? 


5. 22,539,018 +6= ? 


? 


2. 1,621,287--5=f 
4. 39,164,794+6= 
6. 27,622,523 +6= 


? 
? 


130 FIRST STEPS AMONG FIGURES. 
7. 10,763,483+-7=% 8. 45,171,547>7=? 
9. 31,752,200+7=? I0. 508,019,899--8= ? 
11. 34,856,549+-8=? ‘12. 61,476,243—-8= ? 
’ Unless the pupils solve the foregoing examples 
readily, they should review them at once. 


LONG DIVISION. 


The teacher may say to the pupils that when 
the divisor is a large number the method of 
short division is too difficult, illustrating by an 
example. 

Teach the pupils that the first step in solv- 
ing an example in which the divisor is greater 
than 12, is to place a comma after the first 
figure in the divisor as in the example, 
5,02)73245. As in short division we cannot 
divide the whole of a large dividend at once, 
so we cannot in long division. The next step 
is to find how much of the dividend we will 
divide at first. See if the first figure of the 
divisor is less than the first figure of dividend 
or whether it is greater. In the example 
given it is less, (5 being less than 7). Teach 
the pupils that when it is less they are to 
count as many figures in the left of the diyi- 


FIRST STEPS AMONG FIGURES. —13t 


dend as there are figures in the divisor and 
place a comma after the last figure. In the 
given example there are three figures in the 
divisor, so count 3 figures in the left of the 
dividend and writing a comma there, the 
example becomes 5,02)782,45. 

Require the pupils to take these two steps, 
(and no more) with the following examples, 
first placing them in form for dividing: 


76345 +4321. 5738+ 49. 
9458875. 387598858841. 748567+2145. 
57881 ~468. 76854+68. 38768456+25864. 


Teach the pupils that if the first figure of 
the divisor is greater than the first figure of the 
dividend we count one more figure in the divi- 
dend than there are figures in the divisor, in 
order that the part we take may be large enough 
to contain the divisor. * In the example 687531 
—7342 the first figure of the divisor, 7, being 
greater than the first figure of the dividend, 6, 
we count one more figure in the dividend than 
the 4 figures there are in the divisor and the 
example with these two steps taken becomes 
7.342)68753,1. Require the pupils to take 
these 2 steps with the following examples and 

pevee re, 10., D.. Oo. 


132 FIRST STEPS AMONG FIGURES. 
ee ee ee ee 


more if they are needed to make the entire 
class familiar with these steps: 


57281456 — 71235. 
4232145347. 
34216052543. 
684771. 

5875643 +643. 
68475032 +934674. 
47325684+ 3145. 


The next step is to count the number of 
figures at the right of the comma in the divisor 
and count the same number of figures at the 
ieft of the comma in the dividend, and place a 
comma before the one counted which is farthest 
to the left: thus, in the example 34276404, 
the first step is 4,03)34276; the second, 
4,03)3427,6 ; the third, 4,03)34,27,6. 

Require the pupils to take these steps (and 
no more) with the following examples : 


67453275342. 
75694857 +845 321. 
546327 +643. 
345367471 +75382. 
47346 = 23. 
43264726351. 


FIRST STEPS AMONG FIGURES. 133 


8345376-+702. 


56341 68, 
763542 + 8547. 
57345 =, 493: 


In the example 63166 3-+201, which after the 
three steps is 2,01)6,31,663, the next step is, 
see how many times the number at the left of 
the comma in the divisor is contained in the 
number at the left of the first comma in the 
dividend. 2 is contained in 6 three times. 
The example becomes 2,01)6.31,663(3. Next 
multiply. the divisor by this quotient’ figure, 
placing the first figure of the product under the 
figure before the last comma, thus: 

2,01)6,31,663(3 
603 

Next step see if you can subtract. (Teach 
the pupils to look at the left hand of.the num- 
bers to see whether they can subtract. If the 
pupils ask what is to be done when you cannot 
subtract, tell them you will show them in the 
first case in which they cannot subtract, which 
will not occur in the examples given for some 
time.) Next subtract. (Show the pupils that 


the remainder should be less than the divisor.) 


134 _—«¥FIRST STEPS AMONG FIGURES. 


Next see that the remainder is smaller than the 
divisor. 

(Do not show the pupils what to do when the 
remainder is larger than the divisor until a case . 
occurs in their work.) Write the next figure 
of the dividend at the nght of the remainder. 


2,01)6,31,663(3 
603 


286 


Next step count as many figures from the 
right of the partial dividend as there are at the 
right of the comma in the divisor and the ex- 
ample becomes 

2,01)6,31,663(3 
603 


eee 


2,86 


Divide as at first and so continue the opera- 
tion. 

The steps are: 

rst. Write the divisor and dividend in the 
proper form. 


2d. Point off in the divisor. 
3d. Place the right hand comma in the divi- 


dend. 


FIRST STEPS AMONG FIGURES. 135 


4th. Place the left hand comma in the div- 
idend. 

5th. Divide. 

6th. Multiply. 

7th. See if you can subtract. 

8th. Subtract. 

gth. See that the remainder is less than the 
divisor. 

toth. Write the next figure of the dividend. 

t1th. Point off. 

Repeat steps 5, 6, 7, 8, 9, 10 and 11 until the 
example is solved. 

The teacher should solve the following three 
examples zz*h the pupils, before any are 
given them to solvealone. The teacher taking 
the crayon, the pupils will tell what is to be 
done, one pupil describing the first step, 
another the second and so on, or better yet, 
one of them take one of the steps then another 
pupil take another and so on. First solve 
twice the example ANG in the foregoing 
illustration. 


* For method: 


* The pupils should erase the commas which divide 
the number into periods before pointing off, that there 
be no confusion. The commas for the operation -of di- 
viding may be placed above the number instead of 
beneath it, if preferred, 


136 


FIRST STEPS AMONG FIGURES. 


I. 5,450,204 403 = 


2. 
3: 


162,479,845 + 3042= 3 
20,913.844>604=? 


Do not give the pupils more than one or two 
examples each day until you are sure they 
understand the method. 


4. 
. 16,312,418--5,013—=* — Ans. 3,254 
- 273,785,577 + 60,345 = % 

. 26,308,025 4,002-— 

. 189,771,597 +50,364= 4 

. 524,601,734+6,047= 2? 

. 391,838,602 + 80,675 = 

. 619.307,367 + 80,597 = % 

. .278,696,736+6.075 = # 

. 45+3603+456+542 +0 300 aa 


on nui 


4,920,352-+2,023=? Ans. 2,432"° 


116 


454+ 32 +46 +553 +636145= 


- 465+564+324+646+4+ 553 bc ,. 


632 +665'+356+43 +655 +6— F 


. 426 +563 +365 +6344+545 +643 +356 | 


266 +633+56+445+54+63=? 


1. A fox caught 5 geese which were 1-3 of 
the farmer’s flock; how many geese in the 


flock ? 


2. 


A hen had 15 chickens ; a cat caught 4 


of them and a hawk 3. How many were left ? 


See P. Ed., p. 82, 


FIRST STEPS AMONG FIGURES. 1373 


3. Arthur was. paid 14 cents for doing an 
errand ; he lost 5 cents and his sister gave him 
7. How many cents had he then? 

4. A squirrel carried 4 nuts home one day 5 
5 the next day and on the third enough to 
make his number 16. How many did he 
carry home the third day ? 

5. James has 18 apples to divide equally 
among 3 boys; how many shall he give to 
each ? 

6. How many yards of tape at 2 cents a 
yard can I buy for 15 cents and have 1 cent 
left ? 

7. Fred gave 5 cents for an orange, 18 cents 
for figs and 4 cents for a lead pencil. How 
many cents did he spend? 

8. In an orchard the trees.were set 16 in a 
row ;-7 in each row died. How many living 
trees in each row? 

g. A boy sold a pair of doves for 25 cents 
and bought as many marbles at 3 cents apiece 
as he could for the money. How many 
marbles did he get and how many cents left? 

10. A boy earned g cents one day and 12 
cents the next. How much more did he earn 
the second day than the first ? 


c 


138 FIRST STEPS AMONG FIGURES. 


11. Frank had 16 rabbits ; he sold 3 to one 
boy and 4 to another. How many did he 
keep ? 
12. How long will it take a miller to grind 
42 bushels of grain if he grinds 6 bushels in 
an hour ? 

13. Willie bought 2 pass-books at 5 cents 
each ; a lead pencil for 6 cents, and 3 oranges 
at 4 cents a piece. What did he pay for all? 

14. Willie keeps rabbits to sell; he has 20 
and has 4 little houses for them. How many 
does he keep in each house? 

15. If 4 bags contain 8 bushels of grain, how 
many bushels will 9 bags hold? 

16. If 5 cords of wood cost $30, what will 3 
cords cost? 

17. If it cost 15 cents to ride 5 miles on the 
cars, how much will it cost to ride 7 miles ? 

18. How many bushels of oats will 3 horses 
eat in a week, if 6 horses eat 42 bushels ina 
week ? | 
1g. If 6 brooms cost 18 shillings, what will 
5 brooms cost? 

20. If a barrel of flour will last 2 men 6 
months, how long will it last 1 man? 

21. If 2 men consume 6 barrels of flour ina 

See P, Ed., p. 86. 


FIRST STEPS AMONG FIGURES. 139 


certain time, how much will 1 man consume in 
the same time ? 

22. If 4 horses eat 12 bushels of oats in 3 
days, how many bushels will 1 horse eat in the 
same time? 

23. If 3 teams will plow a certain field in 6 
days, in how many days will 1 team plow it? 

24. If 4 men can dig a certain ditch in 8 
days, how long will it take 2 men to dig it? 

25. If 3 men cut 6 cords of wood in a day, 
how much will 9 men cut in a day ?, 

26. If it take 3 men 6 days to cut a pile of 
weod, how long will it take 9 men? 

27. A man lost $6 by selling a cow for $37 ; 
what did the cow cost him? 

28. A boy sold 4 pencils at 2 cents each, and 
3 marbles at 3 cents each ; how much money 
should he receive ? 

29. What is the wheat in 7 bags worth at $2 
a bushel, if there are 2 bushels ineach bag? 

Read the following : 

I. 70000580030. 
2. 68000¢50000. 
3. 680507415371. 
4. 756847597547. 
5. 76500000068. 
6. 67459800009, 


140 FIRST STEPS AMONG FIGURES. 


Teach pupils to write billion. 
1. Write in Arabic, seventy million three 
thousand forty. 
2. Write in Arabic, five billion four mil- 
lion nineteen. 
3. Write in Arabic, ten billion three hun- 
dred million fifty thousand. 
4. Write in Arabic, fifteen ‘billion nine 
thousand. 
5. Write in Arabic, two hundred billion ten 
million twenty. 
6. Write in Arabic, forty million twenty 
thousand. 
7. Write in Arabic, nine hundred forty bil- 
lion one hundred six thousand five hundred. 
8. Write in Arabic, sixteen billion sixteen. 
g. Write in Arabic, five billion forty mil- 
lion. 
to. Write in Arabic, nine hundred billion 
nine. 
11. Write in Arabic, eight billion ninety 
thousand four. 
Teach Roman to 1880. 
12. Write in Roman, one thousand three 
hundred forty-one. 
13. Write in Roman, nine hundred seventy- 
SIX. 
See P, Ed., p. 88. 


FIRST STEPS AMONG FIGURES. I4!I 


14. Write in Arabic and in words MDCCL- 
XXV. 

15. Write in Arabic, ten billion. 

16. Write in Roman, 1876. 

17. Write in Roman, nine hundred forty-nine. 

18. Write in words 761308260017. 

20. Write 6 units of the 8th order, 8 of the 
5th, 3 of the 3d and 5 of the rst. 

21. Express the following number by naming 
the units and their order, beginning at the left : 
70900048010. 


For practice in subtraction, to be given to the 
pupils oradly and recited as the series have 
been. 


CN ie) 


+e (To be read, 45 from 49 ?) as Ne ae 


Moeeo e520 35. 62)..35> 66 19 51. 22 
37.96 45 34.55 29. 65 14 43 19 


45 82 56 32 59 36 51 85 31 56 
AomOOmeAO. 26 ~53 ° 29°40. G6. 25°. 54 


Been to) 35°<72' 4647 , 63.30. 57 
Beeb yre 10552564 43° 38 59 23 54 


Ages 2) 32-04 -46 61 67 48.54 39 
Beee4oez 7 507° 45 54.63 °39- 48 37 


142 FIRST STEPS AMONG FIGURES. 


3L-.70 34 50 24° 51-67) 6h eaeneD 
26 64 25 46 17 48 59 63 79 48 


39-905 
27 86 
For rapid solving. (To be read.) 

I. 3+5+9+7+8+4—6xX9+94+5+7+ 
6—3+7x9—8-8—=3 37 ee 
X8-+9—4—-5-—-7= ee 

2. 657-5 5-3 19%) —-0 7 
X9+ 9+-8+7%9+8-6 27 ee 
—5—5~4x8=? Ans. 64. 

3..61— §-9—5—7+941 > Ocean 
+6—7—8+3 x 8—g—7—9— 5=+7 
%4+7 £4—6= 2 2 Ans, 20 

4. 9x8—6—5—7=90X4-8 X67 Ge 
7+3+9xX6—274xX6—7+7x8+2 
+6X4+4+8=? Ans. 4. 

5. 4X34+8+4x7—-348X5+476X7+ 
8-9 X6+6+5x3+2+4+4+2+648 
+5+o=? Ans. 17. 

6. 5X7+7+6X4+7+7X6—4—5 +3 X 
4+4+8x51+6+6+4x6—3=+5 x3 
+8+5+9=? Ans. 16. 

9. 5X7—-3+8 X64+6+5 x8—3+9x6— 
6+-4xX7—6—8+7X5+6—1+543 
x7—2+6—=? “Ans. 9, 


FIRST STEPS AMONG FIGURES. 143 


8. 4X6+8+8 x9—4+8 x 9—9+34+7= 
eras? ANS, 4. 

9. 6X34+74+44+7+9X44+5+3 x 6—6+ 
6x8+6+9 x 7—8—6+4 x 3+3+8 
mee SO 7 7 ANS: 16, 

10. 44+7+6+94+8—7+34+8+48+5 x6— 
6—3=-4.%8~-7--6—7=-3.xX9+9+5 
Xg9—7—8=? Ans. 66. 

Problems for the slate involving Addition, 
Subtraction and Multiplication : 

1. There are 320 rods in a mile ; how many 
rods in 79 miles? 

2. * William has 75 cents and Charl:s has 
68 cents more than William ; how many cents 
have both boys ? , 

3. David had 123 cents ; he spent 47 cents 
for a ball and 39 cents for marbles. How 
riany cents had he left ? 

4. The larger of two numbers is 916 and 
their difference is 43 ; what is the less number ? 

5. If the drive wheel of a locomotive turn 
around 352 times in going 1 mile, how many 
times will it turn around in going from Canan- 
daigua to Rochester, the distance being 29 
miles ? 


* Tn the different steps of such an example it is im- 
portant as a help to mark each result. 


wee P..Ed,, p. 90, 


144 


FIRST STEPS AMONG FIGURES. 


6. A farmer having 239 sheep, sold 99 of 
them and then bought 113 ; how many had he 


then ? 


7. There are 5280 feet in a mile ; how many 
feet in 357 miles? 


- 3744-645 +57+ 767+4364+543 +675 + 


7+454+765+577+456—=? 


» 454+7764+567+457+7344+475 +674 


7544+5+6764+547+375=? 


- 64347+75+657+746 +773+464+457 


+347+675+766+577=? 


- 5764745 +457+6744+77+556+473 + 


765+7+657+564+705+76=? 


» 5464+375+657+744 765 +257+6+75 


+743+6564+ 7771+467+762=? 


» 647+7564+475 +367+6364+753+77+ 


345 +676+45+576+767 + 654+365=¢ 


. 6.314,532—521,987=? 

- 463,524—39,043= ? 

- 653,425 —64,287=? 

- 475,067 —36,543= 3 

- 688,045 —95,387=? 

- 4,760,352 —376,534= ? 
. 4,630,024—921,045=? 

. 34,000,435 — 2,700,518? 


See P. Ed., p. 92. 


ror 
nun 


= 
Oo ON AM HRW NN SS VN 


i 
fe) 


FIRST STEPS AMONG FIGURES. 


. 4,500,375 —760,187= 2 
‘ 354,000,253 —272,102,437= 4% 
- 530,024—543,052= 3 


. 89,756 xX 96= ? 

- 364,758 x 356=? 

. 638,497 X68=? 

- 498,675 x97=? 
maD.o47 < 70a £ 

. 796,805 X 705=? 

. 807,009 x 608=? 
m0,0590 % 740—> ¢ 

. 94,786 X 7,.968= ? 

© 603,405 —612,134=? 
CT, 


6,53 7,065 aes 743,987 =1 


Division series with remainder, 


g’s and review. 


a | b 
32 15 68 39 61 33 49 26 
5 4,9 8. 7 6:54 
Cc d 
19 37 21 46 78 | 52 53 27 
Mesure 031/77. 8 9 Ox 5° 
| e Brat 
28 54 29 34 61 | 35 58 41 
PGT 288s bet 4G. 7 


145 


146 


+ 
Ha ODO ON AM RW ND * 


= 
N 


FIRST STEPS AMONG FIGURES, 


h 


g 
29 18 40 68 43 21 88 54 25 : 
AG 6 BE 49 8 7 
i 
47 23 34 
6.5 4 


« 33,256,023 +7? 

. 39,044,761+6=? 

- 34,352,839+5= 3 

. 27,832,074-6=? 

- 445,941,095+7= 1 

» 27,516,279+4=? 

. 449,181,483+6=? 

- 487,959,992+7=? 

. 28,556,208 -6=? 

. 15,228,723+4,058=? 

. 251,776,292+70,486= ? 
. 386,124.633+60,578=? 
133 
14. 


41,847,116+9,048—? 
520,613,471 70,697= ? 


Teach the pupils that when any partial div- 
idend (after writing the next figure of the div- 
idend at the right of the remainder) is less than 
the divisor, they must write a cipher in the quo- 
tient, just as they do in short division. Next 


See P. Ed., p. 96. 


FIRST STEPS AMONG FIGURES. 147 


erase the comma made in pointing off the par- 
tial dividend, write the next figure of the divi- 
dend at the right of the partial dividend, point 
off and proceed as before. 

15. 276,265,200+4,036= ? 

16. 42,865,597-+6,075 = ? 

17- 397,706,673 +6,053=? 

18. 44,153,419,619+70,386= ? 

Ig. 15,086,456--4,023=? 

20. 4,437,512,234+60,289= ? 

In the preceding examples the second figure 
from the left of the divisor has in each case 
been a cipher, and the examples have been 
so constructed that the divisor is_ con- 
tained as many times in each partial dividend 
as it appears to be. A new difficulty will arise 
in the following examples since the second 
figure of the divisor is a significant figure. 

Show the pupils that when the second figure 
has value, the divisor is often not contained in 
the partial dividend as many times as it ap- 
pears to be, since in multiplying the divisor by 
the quotient figure there will usually be some- 
thing to add to the product of multiplying the 
first figure, coming from the product of multi 
plying the second. ‘Teach the pupils to observe 


148 FIRST STEPS AMONG FIGURES. 


how much there will be to add to this first 
product and to allow for it. | 

Teach them also that when any product is 
greater than the partial dividend, it shows that 
the quotient figure which gave that product is 
too large, and that the partial product and that 
quotient figure must be erased and a smaller 
quotient figure used. 

Use the following examples in illustration to 
the class, or at least as many examples as will 
make the matter clear to the class: 

267,142+-352=? 
2,659,478 — 461—? 

Use too large a quotient figure in some in- 
stances so as to show the pupils what to do 
when they use one that is too large. Teach 
them to use much care in finding the quotient - 
figure and so save themselves much work. 
Show that the divisor never should be con- 
tained in the partial dividend ro times. 

- 4,367,695 +673=? 
» 220,396--254= ? 
- 307,627+354=? 
. 4,925.151-+-694= ? 
. 6,310,318-+781= ? 
. 3,280,381 + 482= ? 
See P. Ed., p. 98. 


Nm Rh & NN 


7. 
8. 


| FIRST STEPS AMONG FIGURES. 149 


239,294,268+3,642=2? | 
334,990,037 +574= ? 


For rapid solving. (To be read and to be 
answered without use of slate.) 


I. 


To. 


e 


6x6—8+4xX6+8—2+6x4+8+5 x9 
—9+9X6+8+9=2? Ans. 59. 

9xX6—5+7 X6—6+4X7+5+4-9+9 
+64+8—4+3=2? Ans. 9. 


mai O—O— 7-3 X 5+ 7 +6+5--9 X60 +3 


[5xX3+9—7—! Ans. 29. 


» 8X7—9—5+6X 3—-3+3X74+3+5 x 6 


—6—6+6x4=? Ans. 28. 


Ox O—6—7-=-5 X$8444-357X% 449745 


+8+7+5+9=? Ans. 27. 


.4X9+84+4+6 XxX 4—5+3 X6—8 —4+6 


x5+8+4=? Ans. 47. 


7X O—5—9+4x 8—8—6—2'—8 X54+9 


+7—8=? Ans. 33: 


mOx 44-8—-8X%9016—7X% 841-7 xX9—4 


—3-—-7+9——? -Ans. 17. 


- 9X8—6—9—1-7X54+57+9x6+84+4 


pO eOrl yO XA O40. i 5 OX 6 
+7+2+9x8—8—5+9=? Ans. 3. 
7*x*6—6+9+5 x 6—8'-9+5+7 x 8—6 
oo Geeta ok Rg ear Pa Tie mtorr 


150 FIRST STEPS AMONG FIGURES. 


X 4+94+8+9xX7+84+5+8x7=7 
Ans. 42. 

Ir. 9X6—8—7+6+9 X6—6+-3X7+38—7 
+6+7 X4+9+8—7—4+6X8—7 
—6—7+44+74+8+ 8 +4 X74+8=? 
Ans. 64. 

12, 8X84+6+3—7—8—4+6 xX 7—8—6+7 
x 3—8=+6xX090+3—8—45-0 +o oe 
7+6+9—7+5+8—7=2? Ans. 45. 

13. 6xX8+3— 7 —8+9X7+94+8275 x64 
2+7X9—7—9 —8+8 xX 7—8—7+3 
+7+8+7=? Ans. 31. 

14. 7X9—8— 7 +8X9+8—5—6—4—5 = 
6x8+94+97+90+9+ 90 +8 tee 
—7+8X5+8+9+8=? ‘Ans. 60. 


1. How far will a boy walk in 7 days, 
walking g miles each day? 

z. How far will a boy walk in 2 days, 
walking g miles the first day and 7 milés the 
‘second day? 

3. In how many days will a man earn 48 
shillings, at 8 shillings a day? 

4. A boy earned 45 cents Monday, and 53 
cents Tuesday ; how much more did he earn 
Tuesday than Monday ? 

See P, Ed., p. 99. 


FIRST STEPS AMONG FIGURES. 15 

5. If 4 peaches cost 8 cents, what cost 9 
peaches ? | 

6. If a merchant sells 9 spools of thread in 
3 hours, at that rate how many would he sell 
in 1 hour? 

7. If 3 girls can make 6 aprons in a day, 
now many can one girl make in a day? 

8. If 2 girls can do a piece of work in 8 
days, in how many days can 1 girl do it? 

g. A boy bought 9 marbles and lost all of - 
them but 3 ; how many did he lose? 

10. A farmer bought a pig for $6 and sold it 
for $9 ; how much did he gain ? 

11. There were 8 cows in a field and 6 more 
were put in; how many were in the field then ? 

12. There are 17 girls in a class and g boys ; 
how many pupils in the class ? 

13. There are 16 caps in the entry and 7 
bonnets ; how many more caps than bonnets in 
the entry? | 

14. If John is well how many days should 
he come to school in 4 weeks ? 

15. A man may rightly work how many days 
in 3 weeks ? 

16. James has 5 cents and his sister has 2 
cents less than twice as many; how many have 


both ? 


‘ 


152 FIRST STEPS AMONG FIGURES. 


Examples for slate in Addition, Subtraction, 
sda eats and Division: 

. A hasg fields, containing in all 197 acres ; 
B Ny 13 fields, containing 239 acres ; C has 
17 fields, containing 298 acres ; D has 6 fields, 
containing 85 acres; how many fields and 
how many acres have all ? 

2. How many horses at $185 each can be 
bought for 25 cows at $37 each ? 

3. If 69 acres of land cost $6,486 what will 
207 acres cost P 

4. I borrowed of Mr. Rawson at one time 
$697, at another $1,748, and at another $456; 
I paid him $975; how much do IJ still owe 
him P 

5. What is the sum of eighteen thousand 
three, nine million twenty thousand, eight hun 
dred six, seven thousand sixty, 95 thousand 
seven hundred, twenty-one million five hundred 
seventy-six, and ten million ten ? 

6. Henry’s kite was up in the air 375 feet, it 
then fell 98 feet and then rose 268 feet; how 
high was it then? . 

7. Three men bought a hotel for $25, Boon 
the first paid $6,790, the second twice as much, 


See P. Ed., p. 102. 


FIRST STEPS AMONG FIGURES. 153 


and the third the remainder ; how much did 
the third pay ? 

8. The earnings of a father and his 3 sons 
for a year amount to $2175 ; their expenses are 
$957; if the balance is divided equally, how 
much will each have ? 

g. If four dresses of 15 yards each are cut 


from 78 yards of calico, how many yards will 
be left ? 


1. 7+ 46 +744354+47+73+644+57+75 
+64+77+451+63+57+464+75= 

2. 3764+ 455 +757+463+375+747 +654 
+374+576+ 346+775+7+464+ 647 
+356+565=? 

3. 475+647+756+765 +437 +674+575 
+756 +647+ 567+456+743 +357 
+556+463+756=? 

4. 764+354+8+47+ 85 Geg750587-%75 
+84+ 08=? 

§- 58+ 765 —485+ 678 +537+6+753+ 
488 + 846+ 537 +755= ? 

6. 678+845 + 784 + 326+487+856+678 

+588 + 865+478+756=? 

7. 74+58+84+56+78+644-87+6+58+ 
75+86+68+75:=? 


154 


Io. 


II. 


I2. 


Nn WwW NO 


FIRST STEPS AMONG FIGURES. 


. 648+ 785+ 874 +688 +5764+845 +688 


+786 +75 +847+687+58+6=? 


- 7584875 +684+ 768 +475 +886 + 744 


+358 +652+ 887+ 5464785 +648 
eee 

8+57+68+84+75+7+58+76 +88 + 
47+63+78+86+55+67+88=? 

368+ 475 +638+857 +583 +646+878 
57 + 645+ 768+582 +7+6764+3848 
5387+ 766=°? 

67 +788 +856 +475 +687 + 878 + 564 + 
87 +656+478 + 880+ 567 +375 +688 
856+785=? 


- 7,963,034—546,573=? 

. 758,600.341 — 79,423,275 = 4 

. 8,460,075 —987,286= ? 

. 658,000,468 — 35,030,273 = ? 

» 43,750,078 —44,345,621=? Impossible 


6,475,000,374—293,030,596= 


. 97,806 X 59= ? 

- 97,865 x 896=? 

. 96,897 X 6,978=? 

WS 4exX 7, SOO 

. 746,800 X 9,000= ? 
. 470,900 X 70,580= ? 


See P. Ed., p. (10, 


___ _FIRST STEPS AMONG FIGURES. 155 
7. 869,070 X 670,900= ? 
8. 790,600 x 806,700=? 
9. 62,802,889+9=? 

10. 71,262,955—-8=? 

II. 538,908,792+7=? 

12. 376,571,086+-7=? 

13- 339,253,657+9=? 

14. 324,763,528+7=? 

15. 446,217,169+-9=? 

16. 54,284,406+8,047= ? 

17. 25,534,849 +7,197=? 

18. 45,126,612+914=? 

19. 4,368,565 9,168 ? 

20. 63,008,141-+-5,274=? 

21. 47,230,943+-79= 1 

22. 9,290,055,741 + 4,869= ? 

23. 33,279,851 — 48,600 ? 

24. 68,643,216-+87,000 ? 

25. 76,845,678 100= ? 

26. 3,921,534,261-+-486,000= ? 

27. 60,064,175 —8,000= ? 

28. 90,700 X 50,700=? 

29. Subtract 3 billion 6 thousand 750 from 
4s billion 1 million seven hundred sixty-three 
thousand 4 hundred. 

30. 284,553,437+3,790=? 

| See P. Ed., p. 1163 


16 FIRST STEPS AMONG FIGURES. | 


31. 4,167,300,326-+4,790,000= 2 
32. 5,074,000 X 68,070= ? 


33: 3:424,330,021-+ 497,000= ? 
34. 2,468,576,216+10,000—=? 


12’s (and review.) 


For addition and multiplication. 


a b Cc 
9 12 8 Io 7 II ‘9 T2 8 10-7 11 
SiO ulOmEr 1252 O"GgTO rt :12°8. 9 
d e i 
Ee Lote be 7 TAv Ow ie 8 10-9 ai 
T2aaL ibs 2 ae O.1OgL tae 8-9 MOwia 
g h 1 6 
12° 5 12]. 9 12 8 10-7911 |> Ogtgeeee 
Ee Ae 12 6 eae 


12-6650 210 Lia2 


For subtraction. = 0° 

a1 18 20 17320716 

11,(10 O° sos teeme 
d 

20 24.16 19 17 22 

Il 12.6 (Qptowas 


a 
22 19 22 18 Ig 19 
[2 TIZIO/ Oy oes 


Cc 
16 °21023710, 2015 
9-10-11 tO0 Oso 


e 
21° 20/17 20° 23 1874 
12°76 0 $102 2071 


See P. Ed,, p. 140. 


~_—— 


FIRST STEPS AMONG FIGURES. 154 
For division. © aoe 3 Gay 

b 
132 77 80 110 84 48 


21 10 2411 £2: i592 


a 
72 121 108 96 72 I00 
feed t2.°S- 9 10 


Cc 
120 108 72 132 90 99 
Peer. o Tr 10 «=O 


d 
56 88 120 81 96 80 
Seid 2t0). 9:12.°8 
f 


63 110 99 144 64 90 
Ce LO rity ie 725.41 9.7 


e 
60 36 
eat 2 


. 799,896 xX 12=? 
- 6,347,435 X12=? 
- 7,968,473 X11=? 
73,540,247 X 122 
. 989,769 xX 12=? 
- 799,958 xX 12>? 
. 8,989,978 X 12=? 


NI Am BW ND 


12’s (and review.) 
Division with remainders. 


b 
II5 71 96 137 106 85 
10°-9 Toft ye tar 8 
d 
116 102 78 105 86 139 | 124 88 94 95 117 89 
pam Oe LO, Tl) 12°} 10): 9, 8 12°+1I 10 


seein sy Da IL 7. 


a 
127 7694 70 150 108 
migecesgr Oo: 412° 11 


Cc 


S 
115 78 96 130 63 107 
g Sir 12 8 §g 


For more practice. 


For division. 


a 
86 152 40 68 47 
ifs 12a Oley ao 


C 
1627S whl AOI eek 
12S... meaa0 ¢ sOmmeey 


c 
69 54 102 89 45 | 


10 2S rye 


g 
i” O27 OmnsOnG! | 
8 


Goel Tanti2 


8 09.587,888,171 ~i2==? 


gy. 95.62:.647+12= 


Yo. 813,764.564-+-11= ? 


fi. 83,645,840°>13—— 


q 


2 


see P, Ed; 


158 FIRST STEPS AMONG FIGURES. 


b 


78 130 94 62 34 
QT 2125 eee 


d 
52 55 88 64 107 
S : Gari ame 

f 


142 86 52 78 48 
fz tt Gloag, 
h 


57 41 70 107 
ME a «5 


p- 118. 


FIRST STEPS AMONG FIGURES. 159 


For rapid solving. (To be read and to be 

answered without the use of slate.) 

I. 17+4+3X91+8+7+6+12x9—8—6 
yay 6 O+9O+6+9 KI +04847 
179 4 104-34-90+0—24722—1 
Ans. 18. 

2. 144--2+-2--9X8X 2—147X 1243+ 
6-6 X12X2+3+2xX3+9+9-3+ 
2+6+7X4+10X8+4x3+38=? 
Ans. 9. 

3. 9X 12-2+3X2—47+2X3+124+7xX 
12-3 X2>4+1272X31+6+ 3X2 
+775 X8+3X2+3X47+8+9+8 
+9X3+6+9=? Ans, 12. 

Pee 2 + hee] X 12-2 X 3 12 X 1143 
X4+12X8+8—24+373x2+8+6 
BAGS esa a mh 2 T2- 3X 5 
4X 10-42 xX4—4+12=? Ans, 8. 

Re Oe 7 O XS PK 2—- 3X 247 
+7+6 + 2—10X 3—8+5+7xX4xX 
2-415 XQX2X3—9+3X4+11 
x38+3=—2 Ans. 32. 

6. 14X2-4x8+2xX3+7X8+3+2%x4q4 
+82 xX 3—-2+-3X2X2—12xX9+8 
+9—6+10+3~+2+9+3x2+4=? 
Ans. 8. 


ae es 


160 FIRST STEPS AMONG FIGURES. 


7. I2X11+-3+8—+—2+4+8—2 X 3—3+2+4 
X9+3+3X4+24+8+2x343 +12 
X9+-3—274X9+ 3X2>-3X278 
Xa 2s=7) Ansir6, 

8. 19K 24+4+3X 2+4X 124+8+2+245 
2X 3+-2X4~712xK8+2x3+4+8 
X9+134X3+5X2+8 2 eee 

=P Ans 00: 

9. 9X 3—5 +2X343 4 27 90KN6 oe 
+5~+-2xX3+9+8X3+ 2X3+2X3 
~-9X4+2xX 5+2+5xX6+3=? 
Ans. 18. 

10. 16X4—-2—2X3+12+4X3-5X4+2 
X4-2—6>2X5+25x124+8-+ 12 
25 4XI0OS 227 X15 — 50 

3—25-5=? Ans. 5. 

1. If 4 lemons cost 7 cents, what cost 20 

lemons ? | 

Explanation: Teach the pupils that if a cer- 
tain quantity of anything cost a certain amount, 
3 times that quantity will cost 3 times as much; 
4 times that quantity will cost 4 times as much, 
etc. 

Solution: If 4 lemons cost 7 cents, 20 
lemons, which are 5 times 4 lemons, will cost 
5 times 7 cents, or 35 cents. 

See P. Ed., p. 119. 


FIRST STEPS AMONG FIGURES. 161 


2. If 3 oranges cost 10 cents, how many 
oranges may be bought for 30 cents ? 

Explanation: Teach the pupils that if a 
certain sum of money will buy a certain quan- 
tity, 3 times that sum will buy 3 times that 
quantity, etc. 

Solution : If ro cents will buy 3 oranges, for 
30 cents, which are 3 times 10 cents, you can 
buy 3 times 3 oranges, or 9 oranges. 

3. What cost 18 spools of thread at the 
rate of 2 spools for g cents ? . 

4. If 2 knives may be bought for 5 shil- 
lings, what will 20 knives cost ? 

5. If 2 men cut 5 cords of wood in a day, 
how many cords will ro men cut ina day? 

6. If 3 bushels of wheat cost $6, what will 
8 bushels cost? 

7. If 2 bushels of wheat cost $3, how many 
bushels may be bought for $18 ? 

8. 36 cents will buy how many marbles at 
3 for 4 cents ? 

g. If 3 boys can do a certain work in 6 
days, how many days will it take 1 boy to do 
the same work? 

1o. If 2 men can hoe a field of corn in 4 
days, how many days will it take 1 man to 
do it? pte v1 


162 FIRST STEPS AMONG FIGURES, 


11. If 3:men can cradle 6 acres of grain ir 
a day, how many acres can 1 man cradle in a 
day? 

12. If 2 men can build a wall in 6 days, 
how many men can build it in 1 day? 

13. If 4 men can dig a ditch in 12 days. how 
many men can dig it in 1 day? 

14. If 2 men can dig 8 rods of ditch in 1 
day, how many rods can 1 man dig in a day? 

15. If 3 men can dig a ditch in 12 days, how 
many days will it take 4 men? 

Call attention of pupils to the difference 
between the 15th example and the 16th, and 
teach them to find about 1 of the kind the 
question asks about. For instance the 15th 
asks about 4 men, hence find out how many 
days it will take 1 man. In the 16th it asks 
about 6 days, hence find about 1 day? 

Solution of 15th: If 3 men can dig it in 
12 days it will take t man 3 times r2 days, or 
36 days; and four men can dig it in ¢ of 12 
days, or 3 days. 

16. If 3 men can dig a ditch in 12 days, 
how many men can dig it in 6 days? 

Solution: If 3 men can dig it in 12 days, to 
dig it in 1 day, it will take r2 times 3 men or 

See P. Ed., p. 120, 


.< : ‘ 5 i ] 
FIRST STEPS AMONG FIGURES. 163 


36 men ; and to dig it in 6 days it will take 1-6 
of 36 men, or 6 men? 

17. A boy lost 4 marbles, then bought 6, and 
losing 10 he has 35 ; how many had he at first ? 

18. What number multiplied by 3 will give 
race 

19. What number subtracted from 7 will 
leave 4? 

20. How many days will it take 8 men to do 
a work that requires 6 men 12 days? 

21. How many men will do a work in 25 
days that takes 5 men Io days? 

22. What cost 9 suits of clothes at $14 for 
each coat, $2 for each vest, and $4 for each 
pair of pants? 7 

23. How many oranges at 6 cents each can 
be bought for 4 cents and 5 lemons at 4 cents 
each ? 

24. A boy gave 10 marbles worth 7 cents for 
3 figs worth 2 cents each ; how much did he 
lose? , 

25. What cost 60 eggs at 12 cents a dozen? 

26. A boy has 33 cents, how many marbles 
at 3 cents each can he buy and keep 6 cents ? 

27. A boy has 39 cents, how many must he 
earn that he may buy a dozen oranges at 4 
cents each? 


164 FIRST STEPS AMONG FIGURES. 


28. In Mary’s garden are 8 roses, twice as 
many pinks and a dozen daisies; how many 
flowers in her garden? 

29. If 3 pounds of sugar cost 24 cents, what 
will half a pound cost? 

30. Mary has 8 cents, her sister has 6 cents, 
and their brother has half as much as both of 
them ; how many have the three children? 

31. If I buy 60 chickens at the rate of 5 for 
$2, and sell them at the rate of 12 for $5, how 
much will I gain? 

32. How far apart will 2 men be in 7 hours, 
if they start from the same place, and travel in 
opposite directions, one 6 miles an hour and 
the other 4 miles an hour? How far if they 
travel in the same direction ? 

33. A man who drives 9 miles an hour is 
trying to overtake a man who is 24 miles ahead 
of him and who goes 6 miles an hour; in 
how many hours will he overtake him ? © 

34. How many ducks at the rate of 7 for $6 
can I buy for $29 and have $5 left ? 

The pupils should mark each answer, and 
also its denomination. They should be required 
to mark not only the denxomination of each re- 
sult in the process of solving problems, but 


FIRST STEPS AMONG FIGURES. 166 


——— 


what it represents, that is whether it is cost, 
selling price, gain or loss, A’s number, B’s 
number, &c. In this way they will succeed 
with many problems on which they would 
otherwise fail. 


EXAMPLES FOR THE SLATE. 


1. A had $8,948 to which he added $2,284, 
and then he lost $1,632 when he used all he 
had in buying 38 village lots; how much did 
each lot cost ? 

2. B bought 265 acres (of land). for $22,- 
790 ; sold 169 acres of it at $97 an acre and 
the rest at cost. Whole gain? 

3. A horse and 16 oxen are worth $1439 
and the horse is worth $175; what are the 
oxen worth. What is each oxen worth ? 

4. Paid 36 barrels of flour for 60 yards of 
cloth at $6 a yard; how much was the flour a 
barrel ? 

5. If the front and rear walls of a house 
each contain 37,390 bricks, and the other two 
walls each 49,758; how many bricks in the 
four walls ? 

6. If 15 boys walk goo miles in 60 days, 
how far will they walk in 2 days? 

See P. Ed., p. 125. 


166 FIRST STEPS AMONG FIGURES. 


7. Add forty-five million nine thousand 
ten, fifty thousand eight hundred, nine million 
nine hundred thousand seven hundred nine, 
ninety million ninety thousand seven, and six 
hundred seventy-eight. 

8. A sold one horse for $185, another for 
$165, and another for $187; what was the 
average price of a horse? 

g. Divide the product of 6580 and 7900 
by their sum. 

10. A bought 300 acres of western land for 
1,200 ; B bought 275 acres for $175 less, and C 
125 acres at $4 an acre ; how many acres did 
the 3 men buy? How much did they pay ? 

11. A grocer bought 279 pounds of butter at 
27 cents a pound and 98 pounds at 26 cents a 
pound ; he sold the whole at 32 cents a pound ; 
how much did he gain? : 

12. The weight of a number of hogs was as 
follows: 250 pounds, 245 pounds, 260 pounds, 
257 pounds, 273 pounds and 293 pounds ; what © 
was their average weight? 

13. A man wishes to buy a piano for $375 ; 
he lays up $5 a week for a year, or 52 weeks; 
how much more must he save to get the piano ? 

14. Two men start from the same place at 

See P. Ed., p. 128, 


FIRST STEPS AMONG FIGURES. 167 


the same time and travel in the same direction, 
one at the rate of 35 miles a day and the other 
44 miles a day ; how far apart are they at the 
end of 4 days? How far apart if they had 
traveled in opposite directions ? 

15. Divide the product of the sum and differ- 
ence of 364 and 93 by the difference between 
their sum and difference. 

16. A farmer bought one cow for $34, another 
for $43 and another for $61 ; what was the 
average price of the cows? 

17. The product of two numbers is 1,017,702 
and one of them is 2,758; what is the other ? 

18. What is the sum of seventy thousand 
nine, nineteen thousand six hundred forty-nine, 
nine million seven hundred thousand, six 
hundred thousand nine hundred eight, fifty 
million sixty, and three hundred seventy-nine 
thousand eight hundred ninety-eight ? 

1g. The remainder is 713, the quotient 579, 
the divisor 2758 ; what is the dividend ? 

20. A man bought 5 horses at $165 each 
and 6 more for $902 ; what was the average 
price paid? 

21. A woman left her four children $15,000 ; 
the eldest received one-half of it, and the re- 


168 FIRST STEPS AMONG FIGURES. 


mainder was divided equally among the other 
children ; what was the share of each ? 

22. A, Band C sold 20 village lots for $14,- 
600; A received twice as much as B, and B 
$200 more than C, whose share was $3,500 ; 


what did A receive? B receive? 
23. A company of 14 miners sell a mine in 


1,245 shares at $210 per share; what does 
each receive ? 


I. 49 + 7,068 +9,847 +958 +37 + 489 + 8,- 
956 +9843 = ¢ 

2. 946+378+ 795 +849 + 696 + 784 +359 
“+ 4304-775 7.000. 

3- 547+397+ 484+ 758 +969 +847 +958 
+ 497+3844+947+358+596=? 

4. 567+ 498 + 948 + 397 + 846 +372 +458 
+796 + 389+ 486+ 958 +347+598 
=i, 

5. 123+456+789+987 + 456+ 321+743 
+ 398+ 476 +395 +948+767 + 496 
324-1 

6. 578+ 397 +956 + 789 + 437 +4964 875 
+749+ 658 +976+345+876+ 901 
=? 

7- 947 +643 +358+ 895 +769 +5764 348 
+954+ 847 +659 +438 +987 + 648 
320 


FIRST STEPS AMONG FIGURES. 169 


8. 756+395 + 408 + 347+579+943 +658 
+547+ 892 +675 + 487+949 +673 
+246+987=? 

9. 391+849+327 +496 + 327 +843 +659 
+742+869+ 324 +496+932+783 
+468 +-579+453=? 

10. 938+ 493 +745 +679 +548 +987 + 765 
+899+624+ 345 +879+354+497 
+384+947+ 486+ 849+435=? 


PAGE. NO. 


126- 1. 
5,048. 
507. 

- 4,800. 


5,245, 


5,880 
614, 
662. 


. 215,829,875. 
. 196,070,418. 
. 114,326,292. 
. 15,690, 168. 

. 21,269,931. 

. 48,581,904. 

. 53,084,772. 

. 198,177,236. 
. 465,632 13, 

. 364,257 25, 

. 536.174 24, 

. 6,527,465 +8, 
. 3,756.486 26, 
5. 4,603,753 58, 
. 1,537,640 37, 
. 6,453,078 17, 
. 4,536,028 47, 
. 63,502,487 28, 
. 4,357,068 *8, 
. 7,684,530 #8, 
| 13,524 92, 


53,412 541, 


. 34,625 344, 


2,432 118, 
3,254 116, 


» 4,037 312, 


ANSWER. 


PAGE, NO. 


. 6,475 100, 
. 3,768 45, 

. 86,754 296, 
. 4,857 127, 
. 7,684 19, 
12. 
. 4,197. 
. 5,648. 
. 5,049. 
143- 1. 
. 218 cts. 


— he 
OOM 


ee 
Or He Oo 


$00 22D OUR 99 DOH OIE OT G9 D9 


Poi eRe EDUPION. 


ANSWER. 


45,876 36, 


25,280 rds. 


" 10,208 times, 
144- 6. 
. 1,884,960 ft. 


253 sheep. 


5,756. 
5,478, 
5,472. 
6,332. 
6,160. 
7.189 


. 5,792,545. 
. 423,881. 

. 589,138. 

. 438,524. 

. 592,658... 
. 4,383,818. 
. 8,708, 979. 
. 31,299,917. 
. 3,740, 188. 
. 81,897,816. 
. Impossible, 


_ 
DO WIA MP WWE SwoeAH orp gow 


146- 


KEY TO BEEBES 


8,616,576. 
129,853,848. 
43,417, 796. 
48,371,475. 
43,079,676. 
561,747,525, 
490,661,472. 
5,180,582. 
755, 204, 848, 
Impossible, 
5,793,078. 
4,750,860 3-7. 
6,507,460 1°, 
6,870,567 +5, 
4,638,679. 
63,705,870 57, 
6,879,069 3-4, 
74,863,580 36, 
69,708,570 27, 
4,'759, 368. 


. 8,752 3107, 
. 8,572 300, 
. 6,374 461, 
. 4,625 116, 
. 7,364 783, 


15. 68,450 1,000, 


. "2,056 397, 


17. 65,704 361, 


. 627,304 275, 


8,750 208, 


20. 73,604 678, 


. 6,489 598, 
. 867 178, 


869 1. 

7,096 527, 

8,079 619, 

6,805 371, 

65,704 300, 
583,607 219, 

45 fields; 819 A, 
5 horses. 
$19,458. 

$1,926 


. 40,142, 155. 
545 


wf 
153- 8. 


$5,430. 
$304 24, 


9. 18 yds. 
847 


7,600. 
9, 630. 


7,563. 


. 9,066, 


. 1,005, 

. 9,881, 

. 9,687, 

. 7,416,461. 

. 679,177,066, 

. 1,472,789. 

. 622,970,195, 

. Impossible. 

. 6,181,969, 778, 
. 5,770,554, 

. 87,687,040. 

- 676,147,266, 

. 43,050,000. 

. 6,721,200,000. 
. 83,236, 122,000. 
. 583,059,063, 000. 


8.:637, 777,020,000, 


. 6,978,098 7. 
. 8,907,869 *8, 

. 76,986,970 27. 
. 58.795, 869 27," 
. 87,694,850 7°, 
. 46,394,789 ©7, 
. 49,579,685 +, 
. 6,745 7801, 

. 8,547 7,090, 

. 49,372 604, 

. 476 4597, 

. 11,946 4,937, 

. 597,860 3, 

. 1,908,000 3742, 
i 684 37,451, 

. 789 26, 


= 
(Si) 
Ay 


— 
fpane beennerr es 


5. 


FIRST STEPS AMONG FIGURES. 173 


. 768,456 78, 
_ 8,069 261, 
7,508 275, 


4,598, 490,000. 


- 42,001,756, 650. 
0. 75,080 237, 


870 326 


32. 345,387,180,000. 
_ 6,890 21, 

| 246,857 6216, 

. 9,598,752. 
76,169,220. 

. 87,653,208. 

. 42.554.964. 


11,877,228, 
9,599,496. 


‘ 107, 879, 736. 
798, 990, "680 11712, 


7,968,470 7-12, 
73.978,596 11, 
6,970,486 &12, 
$252 24-38, 
$1,859. 


: $1, 264; $79. 
$10 


174, 296 bricks. 
30 mi. 


166- 7. 


167-15. 


145,051, 204. 
$179, 

if 3,589 13,280-14,480 
. 700 A.; $2,725, 
» 1,983. cts. 

. 263 tbs. 

$115. 


. 86 mi; 316 mi. 
665 157-186, 


‘ S157 [ $2, 500 each. 


21. Eldest $7,500; oth’s 


169- 


10. 


$0 90's > OUP CO BO r= ¢ 


. A, $7,400; B, $3,700, 
$18,675. 


eet 
er 
= 


ENpD OF KEY TO TEACHERS’ EDITION, 


PUPILS’ EDITTORs 


PAGE. NO. ANSWER. PAGE, NO. ANSWER. 

33-1. 251. 34. 317. 

2, 272. 37-35. 21.752. 

8. Q71. 36. 24,853. 

4, 239, | 37. 63,545, 

5. 234, 38. 22.544, 

6. 239. 39. 93,733. 

7, 262. 40. 62,592. 

8. 324. 41. 66,651. 

9, 252, 42. 77,444, 

10. 260. 43. 28.345. 

34-11. 296. 44, 72,353. 

12, 295. 45, 62,156. 

13. 308. 46. 63,234. 

14, 335. 1. 46,206. 

15. 314. 2. 90,693. 

16. 325. 3. 39,606, 

17. 313. 4, 42,064. 

18. 327. 38-5. 93,069. 

19, 324. 6. 24,640. 

35-20. 316. 7. 60,369. 

91. 317. 8. 69,396. 

22. 346. 9. 62,840. 

23, 319. 10. 96,908. 

24. 302, 11. 60,846. 

95, 845. 12. 69,369. 

26. 307. 13. 32,341. 

97, 207. 14. 31,208. 

36-28. 325. 15. 10,243. 

29. 328. 16. 20,413. 

30. 308. 17. 18,028. 

31, 293. 18. 41,302. 

32. 323. 19. 14,032. 


33. 303. 20. 80,218. 


FIRST STEPS AMONG FIGURES. 


. 13,024. 
. 40,132. 
. 20,312. 
. 2,429, 
. 2,463. 
» 2,544. 
. 2,420. 
. 2,349. 
. 2.587, 
2,917. 
* 21996. 
2. 2,364. 
_ 2,967. 
. 3,178 

. 3,184. 

. 34,892. 
. 36,775. 
. 85,561. 
. 35,673. 
. 85,680. 
. 4,655. 
4.514, 
4.615. 
. 4,300. 
A522. 
. 4,837. 
. 49,044. 
. 48,573. 
. 48.874. 
. 49.083. 
. 51.204. 
. 4.258. 
4.011. 
. 4,741, 
4,743. 
. 4,893. 


3 

4 

5 
6. 
48-7. 
8 

9 

10 


. 563,135. 
. 741,027: 
. 411,303. 
. 623,024. 
. 71,634. 
. 44,252. 
. 443,043. 
2. 476,446. 
. 556,467, 
. 375,667, 
. 77,567. 
. 766,475. 
. 876,737. 
. 875,565. 
. 666,567, 
TUT 657. 
. 267,146, 
. 365,662, 
S80T TT: 
. 176,515. 
. 262,607, 
. 421,256. 
. 277,465, 
. 219,173. 
. 373,676. 
. 378,074. 
| 257,156, 
. 471.465, 
BLTTTS, 
. 76,516, 
. 311,856. 
. 677,172, 
- 376,046. 
. 562,567, 
. 40,895, 
. 42,073, 
- 41,217, 


44,286, 
42.308, 
44,509, 
41,808, 


. 46,765, 
- 46,558, 


48,802. 


- 4,628, 


175 


176 KEY TO BEEBE’S 
12. 8,462 2. 73,276, 956. 
13. 4,028 3. 79,097,578. 
14. 60,248 4, 71,178,967. 
15. 63,906 5. 87,819,971. 
16. 68,402 6. 3,765,687. 
17. 46,082 7. 68,797,784. 
18. 70,492 8. 5,748,957. 
19. 127,068 9. 73,287,865. 
20. 92,704 10. 86,995,768. 
21. 106,926 11, 80,773,478, 
22. 109,356 12. 26,579.678. 
23. 79,359 65-13. 7,170,658. 
24. 92,704 14, 94,880,045. 
25. 193,578 15. 860,697,867. 
26. 214,496 16. 47,159,685. 
27, 198,572 17. 32,660,068. 
53-28. 213,704 18. 6,697,868. 
29, 258,144 19. 77,995,682. 
30. 71,284 20. 90,859,567. 
31. 139,056 21. 77,706,580. 
32. 145,704 22. 93°730,145. 
33. 105,738 23. 35,764,946. 
34. 218,712 24 70,299,847. 
35. 219,276 25. 77,660,078. 
36. 387,156 67- 1. 246,722. 
37. 231,760 2. 296,492 
38. 254,096 3. 213,852 
39. 140,168 4, 492,241 
40. 202,680 5. 249,494 
41. 182,772 6. 281,968 
42. 230,17 7. 459,704 
43. 157,824 8. 211.476. 
44. 150,216 9. 368,464. 
45. 312,384 68-10. 5,480,296. 
59- 1. 52,279 11. 3,395,469. 
2. 45,897 12. 5 142,276. 
3. 57,437 13, 2,548,282, 
4. 50,384 14. 2,548,546. 
5. 51,459 15. 29,126,064. 
60- 6. 47,931 16. 53,085,480. 
7. 54,123 17. 1,483,264. 
8. 54,794 18. 1,295,866. 
9. 57,547 19. 84,320. 
10. 56,480. 20. 15,674,618. 
64- 1. 7,867,676. 21. 1,586,880. 


aoslet adalat atts 


FIRST STEPS AMONG FIGURES. 


. 2.298.402. 
. 24,593,992. 
. 41,303,168. 
. 4,828,052. 
. 313,225. 
”, 4,252,262. 
. 22,517,838. 
. 3,186,721. 
. 3,430,458. 
. 11,993,364. 
2. 27,078,591. 
. 47,923, 624. 
. 2,011,205. 
. 16,799,022. 


. 53,208,409. 
. 2457 ¥8, 

. 3,543 14, 

5 ATB 23, 

. 274,658 3, 

. B45, 734 84, 

. 265.435 *5, 

. 238,636 24. 

. 1,887,677 ©. 
. 6,543,346 *, 


54,764 35, 


. 154,264 46, 


64,357 14. 
623,542 26, 
75,246 *5, 
543,452 47. 
974,658. 
642,455 &7, 
646,819 2. 
786.538 "4, 


. 24,635 37. 

. 3,456,827 *¢, 
. 63,287. 

. 61,205. 

, 63,924. 

. 66,698. 

5. 65,307. 

. 55,344. 

. 52,771. 

. 57,783. 

. 61,074. 

. 60,493. 

. 69,274. 

. 69,883. 

. 68,111. 

. 71,276. 

. 67,740. 

mags 438, 656. 
naa: 971, 752. 

; 244,806,912. 
. 382,698,325. 
. 23,698,382. 


26,155,500. 


. 2,595,136. 
. 4,188,375, 
. 5,965,938. 


3,235,848. 


. 55,044,555, 

- 17,508,384. 

. 26,435 *4, 

. 635,241 25, 

. 854,624 14, 

. 8,746,254 34, 
. 657,342 1-4, 

: 4,635,246 +, 
. 7,562,435 24, 


20. 437,526 46, 


rw) 
oO 


. 4,537,264 37, 
2. 5,743,263 **, 
. 5,762,474 &6, 
. 2,605 4, 
6,370 524 #7, 
~ 423,177. 

. 6,258,085. 

. 3,604,756 *7. 


177 


KEY TO BEEBE’S 


- 4,653,702 &8, 
- 7,086,534 26, 
. 5,768,430 47. 
- 6,573,048 38, 
. 2,540,867 &8, 
. 4,352 143, 

. 3,624 152, 


4,235 214, 


- 4,236 1,022, 


5,342 647, 
24,353 1408, 
G45 1125, 


. 3,624 231, 

. 3,425 240, 
aul Oa 

. 3,426 2,120, 
. 3,564 3182, 
. 6,457 341, 

. 54,673 1,342, 
2) 4. TG: 9, 0005 
. 57,643 216, 
. 3,745 198, 

. 4,576. 

. 74,656 991, 
. 46,576 397, 
. 6 534 279, 

- 76,487 1329, 
. 7,458 8. 


8. 6: 47, "664 2.388, 


. 174 marbles. 
. 632 bu. 
. 189. 


© 

@ 
oS ! 
SD WED OUP G9 2 
[@r) 


. $12,250. 

. 127 marbles. 

. 105 cts. 

. 1,781 steps. 
$85 


. 7,682 cts. 
. $362. 

. 92 da. 

. 425 

. $668. 

. 91 marbles, 
. $1,920. 

. 25,092 cts. 
. 59 bu. 

. 15 ets. 

2. 874 bu. 

3. 152 bu. 

. 1,784,910. 


64,482, 


94-11.3,608, 157. 
12. 658,280. 


. 616,178. 

: 266.078. 

. 6,050 993. 
. 391.648. 

. 407,708. 

. 278,079. 

. 277,164. 

. 391,667. 

. 622,678. 
2. 578,238. 

. 39, 213, 566. 
. 670, Tal. 

. 3,764,877. 
. 79,088. 

. 24,147, 


FIRST STEPS AMONG FIGURES, 179 


28. 42,869,909. H 12 5,746,387 +7, 
29. 69,059,978. 13. 7,468,576 7°. 
30. 37,523 976. 14. 67,580,760 ©. 
31. 3,189,825. 15. 4,637,586 7-8. 
2. 36,970.172. 16. 65.748,760 37. 
1. 70 919.205. 17... 796,859,809 *¢, 
2. 47,009,130. 18. 697,879,680 3-7. 
95- 3. 5,449 025. 19. 64,859 760. 
4. 302,421,261. 20. 748,695 47. 
5. 31,269.744 21. 479,568 !- 7, 
G. 392. 262,525. 97-22. 840,967 *°8. 
7. 3,756,608. 23. 639,408 7-8, 
8. 726.525. 24, 4,253 526, 
9. 187,044. 25. Omit this example. 
10. 7,221,816. 26. 35,246 108, 
11. 5,498,265. 27. 5,264 2243, 
12. 5,448, 946. 28. 6,423 1-243, 
13. 4,231,810. 29. $63 12,217, 
14. 35,677,072. 30. 4,536 1036, 
15. 64,367,975. 31. 64,352 341, 
16. 49,849,081. 32. 48,372 345, 
17. 610,079,611. 33. 57,362 3.782, 
18. 308,940,676. 34. 41,572 395, 
19. 28,681,464. 35. 3, 754. 1,000, 
20. 379,544,763. 36. 4,827 314, 
21. 35 695,072. 37. 36,472 247, 
22. 68,097,323. 38. 64 727 4.6%, 
23. 58,390,182. 39. 468,752 4,306, 
24. 379,239,951. 40. 74,635 325, 
25. 6,257,507,544, 41. 4 605 325, 
26. 6,328,767. 42, 70,534 3-041, 
96-27. 2,816,929. 43. 4,653 1124, 
28. 708,709. 44. 64,075 38, 
29. Impossible. 45. 736,502 128, 


463,057 37, 98-46. 475,630 2345, 
634,750 *8. 47. 4,607 3,762, 
475.307 3°, 48. 60,835 31, 
5,463,060 +9, 49. 72,506 119, 
564,037 29, 50. 43,072 3.060, 
374,675 &9, 51. 4,073 19-301, 
eres 3-8, oy 47,260 479, 
. 6.5 rye-e . 58,240 749, 
6,870,657 &7, 54. 47,050 78, 
4, 857,680 69, 55. 70,648 2876, 
. 6,578, "648 28, 56. 4,756 127, 


“EH SOM MMO mR ONe 


© fmt pe 


180 


KEY TO BEEBE’S 


« 697 247, 
10S 457, 
- 579 433, 


196 527, 


Reed Oued 

“1 108 143, 

« 687 541, 

» 80,743; 

. 8,607 801, 
. 6,908.74. 
» 8.097, 47°, 
. 7,906 473, 

. 47,650 230, 
- 76,305 2°, 
. 67,057 9.246, 
2. 57,642 787, 


if 592. 


109 sheep. 


. Lost $900. 
. 1,796,256 apples. 


$4,543. 


. 59,888,884. 
. 136 tons, 
8. 337,022. 


$19. 


20. 191,284 yd. 


; Lost $50. 
. 14 horses an 
3 


| 421 A.; $37,309. 
fs $6 54-135, 


6. 
105-27. 
28. 
29. 


rem. 
$50 


$4,456. 
3,743,520 ft. 
33 mi, 
Nothing. 


30. 
dl. 
106- 1. 


86 yrs. 
840,413,978. 


; 389, 385, 828, 
“i bs 165, 668. 
29.730,808.. 
. 7,080,807. 

. 291,864,906. 
. 77,927,921. 
. 730,868,068. 
. 20,679,828. 
. 7,779,807... 

. Impossible. 
. 89,299,876. 
- 4214769,927. 
. 670,796,919. 
. 6,479,979, 692. 
. 4,585,944. 


; 772, 926, 042. 
‘379,239,951. 
| 550,842 699. 
6,953, 140,080. 


113- 


FIRST STEPS AMONG FIGURES, 


. 490,619,928. 
5. 4,311,281,464. 
78,650,000. 


’ 


. 6,800. 

. 6,320,000. 

. 87,500,000. 

. 172,500,000. 

. 16,205,000. 

. 444,500,000. 

. 18,700,000,000. 
. 4,336,400,000. 
. 17,983.00. 

. 51,300,000. 

. 6,887,200. 

. 59,073,000. * 

. 40,843,000. 

. 8,407,454,000. 
. 452,276,640, 000. 
. 3,589,638,200,000. 
. 4,827,581,000. 
. 6,409,800,000. 
. 4,798,697 5-9, 

. 87,896,790 7%, 
. 769,589,079 *8, 


9.687,890 2-7, 


- 769 809,780 &9, 
. 6,589,679 5-8, 

- 978,697,089 6-9, 
. 76,898,697 4-6. 
. 7,049,680 5-7, 

. 4,906,704 7-9, 

. 79,684,796 4-7. 
- 958,007,980 8-8. 
. 893,798,400 7-9. 
. 65,870,486 4-7. 
. 870,956 5-6, 

- 75,680,390 57, 


903,780 5-8, 


- 9,586,090 7-2. 
. 89,607,980 3-8, 
. 5,869.759 +7, 
» 4,375 6001, 
5,736 621, 

865 216, 


116- 


OUP 6929 


. 147,283 7.069, 
. 6,485 70°. 

. 46,372 528, 

. 14,735 499, 

. 17,399 321, 

. 68,591 7457, 
- 147,964 948, 
. 9,608 4579. 
2. 13,568,274, 

. 49,807 291, 

. 138,769 3,700, 


79 3,468, 
376 8°, 

23 47,600, 
762,196 48, 
OL 2,938,641, 
486 3211, 
6,857 12,131, 
796 75.300, 
96 21 352, 
873 4758, 
98 30 000, 
468 2,300, 
2,005,600,000. 
7,958 123, 
68,050 759, 


. 7,960 2°. 
_ 5,920,005,998. 
. 48,690 20,172, 

- 79,080 790, 
1,785 76,495, 

6 60,970 28. 

. 9,780 58, 


Impossible. 


i 740 157,000, 

_ 8,069 37,281, 
. 6,095 2°, 

. 405,025,600. 
. 78,096 231, 


796,000. 


. 947,604. 


767,679. 


. 9,058,032. 
. 10,149,216. 


749,042,756. 


181 


KEY TO BEEBE’S 


. 7,043,688. 


78,956,244. 
833,855,616. 
104,154,369. 


. 5,755,896. 


1,171,075, 764. 


2. 83.710.764. 

. 837,443,940. 

. 1,136. 156,268. 
. 11,758,752. 

. 7,543,756. 

. 105,587 
. 598,772,364. 

. 1,175.855,832. 
. 95 879,820. 

. 98, 
. 49,688,132 92, 
- 68,374,989 #12, 


748, 


790,948 1-12, 


7,076, 859 7-12, 


"786,547,997 922, 
. 869,897,046 311, 
. 37,998,060 1012, 
. 79,684,968 92, 
. 74,869,740 512, 


30. 675,846,090 211 


125-1. : 
. $34,995. 
- 490,000. 


8,552 
$760. 
. $14,352. [& $27 left. 
1742797A.; or 174 A. 


. 614 mi. 
: 22,999, 800,925. 
2,250. 


$2,010 


17 yrs. 
236 A.; $74 136.236, 
15, 148. 


: 3.066. 


ius 


. 22,032 solid ft. 
. $4,340. 
» $8,380. 


129-29. 
0. 69 horses, an 
. 42 times, 


133 


21. 
22. 
23. 4 
24, 
25. 
26. 
au. 


28. $45 


118 79-158, rem. 


$53 


. 398 lbs. 
130-38. 
. 49 animals. 
. 6 yrs. 
. 1,440 Ibs. 
2. 3,536. 

. $772. ae 
.6 horses anc 
. 1,274 bags. 
131-46. 


15 sheep. 


$40 
32 teachers & $175 


remaining, 


. 14 cows. 

. 949 cts. 

. Gained 80 cts. 
. 70 half dimes. 
. 87 bags ; 
2. 1382 birds. 
132-53. 
54, 
55. 4 


518 cts. 


1,162,568 880-1,090 sec, 
$4,730. 
8. 


Omit ** more.” 


56. 
. $86. 

. 6,784. 
. 76 Co’s and 15 men 


. af yds. 

. 1 215rdez 9,051 ra 
. 12 yrs. 

. 158, 


Lost $685. 
[left. 


FIRST STEPS AMONG FIGURES. 


64. 8,589,783. [ 902. 10. 74,617. 
65. Quo. 105; rem. 3,- | 13'7-11. 90,190. 
66. 2,392. 12. 95,621. 
135-1. 70, 169. 13. 93,522. 
2. 67,005. 14. 89,651. 
3. 58,369. 15. 93,822. 
4, 74,595. 138-16. 104,643. 
5. 74,473. 17. 168,075. 
136- 6. 76,967. 18. 98,259. 
7. 81,025. 139-19. 111,157. 
8. 73,892. 20. 111,236. 
9. 72,772. 


APPENDIX. 


DETAILED METHODS IN ARITH- 
METIC. 


FROM THE COURSE OF STUDY PREPARED FOR THE PUB- 
LIC SCHOOLS OF SAN FRANCISCO. 


I. Lessons FOR BEGINNERS. 
Grube’s Method in Number. 


THE following are in substance some of the 
most important principles given by Grube for 
his method in teaching beginners to compre- 
hend numbers and their relations. 


Principles. 


““r. Each lesson in Arithmetic must also 
be a lesson in language. The teacher must 
insist on readiness and correctness of expres- 
sion. As long as the language for the number 
is imperfect, the idea of the number will be 
defective. 

“2. The teacher must require the scholar 
to speak as much as possible. 

‘3. Answers should be given occasionally 

185 


186 APPENDIX. 


by the class in concert, but usually by the 
scholar, individually. 

“4. Every process must be illustrated by 
means of objects. 

‘““s. Measure each new number with the 
preceding ones. 

“6. Teachers must insist on neatness in 
making figures.” 


ORDER OF STEPS. 


first Step. Ulustrate the required combi- 
nations by means of counters, such as blocks, 
splints, or shells, in the hands of the children 
themselves, and by other objects in the hands 
of the teacher. 3 

Second Step. Express the. same combina-’ 
tions on the blackboard or on slates with 
marks, 

Third Step. Take the same combinations 
mentally with abstract numbers. 

Fourth Step. Practical problems in applied 
numbers. 


HOW TO BEGIN. 


[=s~ The time required for the work will 
depend upon the age of the children, as also 


APPENDIX. 184 


somewhat upon their natural ability. Some 
children may require a year to complete the 
work which others may master in a term. 


RIae NOMBER ONE. 


1. Hold up one counter, one hand, one 
finger, one slate, etc. 

On your slate make a straight mark, one 
dot, one cross, etc. 

On the blackboards make one mark, one 
dot, one cross, etc. 

2. Place one counter in the middle of the 
desk; take it away; how many have you 
left ? 

Make one mark on your slate ; rub it out ; 
how many marks are left ? 

3. Send the class to the blackboards and 
let them make the mark for one thus, | ; and 
also the figure thus, 1. 

4. Proceed very slowly. Much time should 
be given to those who do not learn easily. 


Il. THE NUMBER TWO. 


1. Each of you take one counter and place 
it by itself on your desk ; now take another, 


188 APPENDIX. 


= wy 


and place close to it ; how many counters 
have you? (Require the answer in a full sen- 
tence.) 

Make one straight: mark on your slate; 
make another close to it; how many have 
you now? 

Go to the blackboard : make one mark ; 
another, close to it ; how many now ? 

Clap your hands once; again; how many 
claps ? 

Rap on your desk once ; again; how many 
raps? 

2. Counting —Place one counter on your 
desk,* ; a little way off from the first one, 
place two counters close together, thus* *. 
Count, one two ; two, one. 

On your slates make marks thus, | ||, and 
count forwards and backwards. 

3. Addition.—I. Place one counter on the 
desk ; place another counter close to it; how 
many have you now? <Anzs. J have two count- 
ers. Howmany counters are one counter and 
one counter? Ams. One counter and one 
counter are two counters. |The teacher will 
further illustrate with books, pencils, crayons, 
etc. | 


APPENDIX. 189 


Il. Slate and Blackboard.—Make one mark ; 
another one near it. How many marks have 
you made? 

[Continue with rings, dots, crosses, etc. | 
4. Subtraction.—Il. Place two counters to- 
gether on your desk; take one away; how 
many have you left? Ans. [have one left. 
One counter from two counters leaves how 
many ?. Ans. One counter from two counters 
leaves one counter. 

| Teachers will continue with fingers, hands, 
books, and other objects. | 

Il. Slate and Blackboard.— Make two 
marks; rub out one; how many are left? 
Make two marks; rub them out; how many 
are left? Ans. Wone are left. Two taken 
away from two leaves how many ? 

5. Multipliication.—I. Each of you put one 
counter on the desk ; now put another one 
with it ; how many times have you taken one 
counter? Ans. [ have taken one counter twice. 
Two times one counter are how many count- 
ers? Ans. Twice one counter are two counters. 

Il. Slate and Blackboard—Make one mark ; 
now another. How many times have you 
made one mark? Azs. [have made one mark 


190 APPENDIX. 


twice. ‘Then two times one mark are how many 
marks? Ans. Two times one mark are two 
marks. 


6. Diviston.—See Teacher’s Edition of 
First Steps, page 14, paragraph to. 


7. Compartson.—Give one counter to John 
and two to Frank. How many counters has 
John? Frank? How many has Frank more 
than John ? How many more is two than one ? 

How many counters has John less than 
Frank ? Then one is one less than two, and 
two less than two is nothing. 

Blackboards. —lustrate the same with 
marks. | 


General Remarks. 


Itisa feature of this method, that it teaches 
by the eye as well as by the ear, while in 
most other methods arithmetic is taught by 
the ear alone. If a child is to measure 7 by 
the number 3, the illustration, by comparison 


Sis 
% 


APPENDIX. 191 


“Tf counters are arranged in this way, and 
impressed upon the child’s memory as depict- 
ing the relation between the number 3 and 7, 
it is, in fact, all there is to know about it. 
Instead of teaching all the variety of possible 
combinations between 3 and 7, it is sufficient 
to make the child keep in mind the above 
picture. The first four rules, as far as 3 and 
7 are concerned, are contained in it, and will 
result from expressing the same thing in dif- 
ferent words, or describing the picture in dif- 
ferent ways. Looking at the picture, the 
child can describe it as: 


34+3+1=7, 0r3 x 2+1=7, PE D3 = Sar 2, 
473-2 (1). The latter process is to be 
read : 3 in 7 twice, and 1 remaining. 

“Let the number to be measured be 1o, 
and the number by which it is to be measured 
be 4; then the way to arrange the dots is: 


* * * * 
* * * 
* % 


“The child will be able to see at once, by 
reading, as it were, that 4+4+2=I10, 4x2 
+2=I10, lo—4—4=2, 10+4=2 (2), and to 


192 APPENDIX. 


perceive at a glance a variety of other com- 
binations. The children will, in the course 
of time, learn how to draw these pictures on 
their slates in the proper way. Nor will it 
take long to make them understand that every 
picture of this kind is to be ‘read’ in four 
ways, first using the word avd, then émes, then 
fess, then zz. As soon as the pupils do this, 
they have mastered the method, and can 
work independently all the problems, within 
the given number, which are required in meas- 
uring.” 


Order of Steps. 


I. Counters. 
II. Figures. 
III. Abstract Numbers. 
IV. Practical Problems. 


FIRST STEPS 


i 


AMONG » Ficures. 


it 
A Drill Book in the er ai Rules 


of Arithmetic 
( 
wn qi 
Be? Berrien. ie 


nv t) 


f BY, / 
LEVIRN. BEEBE,» 


Je, 
oe ae (EANANDAIGUA, N. Y. 


Be caae N. Y.: 
C. W. BARDEEN, PUBLISHER. 
1884. 


Copyaicat, 1877, Levi N. Berse. 


¢ a : 
x i 
- —_———__ -— 
’ 
. 
A 
¢ 
s 
on 
‘ 
- 
— 
* 


PREFACE TO REVISED EDITION. 


Great care has been taken to correct the 
errors of the first edition and it is hoped that 
few remain. 

At the request of several teachers in un- 
graded schools, the addition, subtraction, mul- 
tiplication, and division tables have been in- 
serted in the Pupils’ Edition, and a number of 
pages have been prefixed to relieve teachers in 
‘such schools of much of the labor of oral in- 
struction. In graded schools the pupils should 
not have a book until they have been taught 
orally as far as to the tables of 7’s and review, 

mag: 

The Teachers’ Edition contains the answers 
to the examples in this book, and instruction 
for oral work with the youngest pupils, together 
with methods and additional examples. It is, 
therefore, necessary that the teacher should 
have it and carry along the work of the two 
editions together. References at the bottom 
of the pages in each edition call attention to 
the pages of the other edition which contain 
work of the same kind. 

The Pupils’ Edition is bound separately for 


iv PREFACE TO REVISED EDITION. 


pupils’.use, as well as with the Teachers’ Edi- 
tion for teachers’ use. . 

The object of having the double book, the 
Teachers’ Edition and the Pupils Edition, is. 
that while the pupil is to prepare his lesson 
from the Pupils’ Edition, the teacher has, in the 
Teachers’ Edition, additional examples which 
the pupil has not seen, which are intended to: 
be assigned for solution during recitation as a 
test of his knowledge of the subject in hand. 

The first g6 pages of the Teachers’ Edition, 
together with the parallel work of the Pupils” 
Edition to p. 44, are bound separately for the 
use of teachers in the oral instruction of pupils 
during their first two or three years in graded. 
schools. It is intended wholly for oral work 
and is called “ First Steps Among Figures, Oral 
Edition.” 

Much care has been taken in each edition to: 
proceed from the easiest examples to those that 
are more difficult, in order to avoid discourag- 
ing the pupil. 

The author is indebted to his assistant teach- 
ers for aid in the preparation of the very large 
number of examples in the book. 

For a more extended notice of the scope and: 
plan of this work see the preface to the Teach- 
ers’ Edition, and also the Special Notice which. 


precedes it. 
LEVI NieeB iio 
CANANDAIGUA, N. Y., April, 1878. 


Seti NCTC EK: 


Persons who may examine this book are 
asked to notice especially the following: Ex- 
amples for rapid solving on pp. 17, 20, 25, 29, 
38, 41, 45, 51, etc., as well as the foot notes, 
pp. 16 and 17; the drill in reading and writing 
mumpbers On pp. 39, 47, 53, 60, 69, 33, 111 to 
116, etc.; the series of division with remain- 
‘ders, which is to prepare the pupil for short 
division, pp. 107, 128, 129, 145, and 157.. In 
the examples in long division, on p. 136, Teach- 
ers’ Edition, and on p. 82 of the Pupils’ Edi- 
tion, since the second figure from the left in the 
‘divisor is a cipher, while the figures of the 
quotient are small, the divisor is contained in 
each partial dividend just as many times as it 
appears to be. For instance, the first divisor 
in the Pupils’ Edition, p. 81, is 201 ; and if the 
reader doubts that the path is thus made easy 
for the beginner, let him give his pupils an ex- 
ample with 19, 29, or 291 for a divisor, and 
then one with zor asa divisor. There are no 
‘ciphers in the quotients on those pages, so that 
every difficulty is postponed to a later time that 
can beso put off. The examples were made 


vl SPECIAL NOTICE. 


by assuming such a divisor and quotient as 
were desirable, multiplying them, and adding 
an assumed remainder to the product, which: 
gave the dividend found in the book. 

After the practice on the twelve examples on 
p. 136 in the Teachers’ Edition, and on the 
twenty-nine examples on p. 82 of the Pupils” 
Edition, there are a number of pages of other 
work, after which long division, with the same 
sort of easy examples, recurs on p. 146, Teach- 
ers’ Edition, and p. 97, Pupils’ Edition. After 
a few examples the cipher occurs in the quo- 
tient, of which the teacher is warned at the 
bottom of p. 146, Teachers’ Edition. Addi- 
tional difficulties are treated on pp. 147 and 
148, followed by examples in illustration, and 
still others on pp. 114, 115, and 116, Pupils” 
Edition. 

The continuous form commonly used for 
tables of addition, subtraction, multiplication,. 
and division, (as 2 and 2 are 4, 3 and 2 are 5, 
4 and 2 are 6, etc.), has been forsaken for the 
form found on pp. 33,.37;43, 49; 50; 03, cram 
Teachers’ Edition, and on pp. 6, 9, 18, 24, 29,. 
etc., Pupils’ Edition. 

If the teacher copies the series on the black- 
board, he may write the answers underneath, 
or require the pupils to find the answers, as he’ 
prefers. 

For instructions see pp. 42 and 43, Teachers’ 
Edition, and pp. 5, 6, and 1g, Pupils’ Edition. 

The teacher will see that at p. 63 he should: 
begin to use also the Pupils’ Edition, even: 


SPECIAL NOTICE. vu 


though the pupil has not yet obtained his book. 
The number at the bottom of that page refers 
to the page of the Pupils’ Edition that has the 
same kind of work. ‘The numbers at the bot- 
tom of the following pages in the ‘Teachers’ 
Edition refer in the same way, while those in 
the Pupils’ Edition refer back to the Teachers’ 
Edition. The work of the two editions after 
reaching these parallel pages should be carried 
along together carefully. 

The operations of addition, subtraction, mul- 
tiplication, and division being taught together 
throughout the whole book, beginning with the 
easiest examples and progressing gradually to 
more difficult ones, work in each rule is con- 
stantly recurring. ‘This necessitates a peculiar 
arrangement of the work, but it constitutes one 
of its chief excellencies. 

The Pupils’ Edition is bound separately for 
pupils’ use. 

The key containing answers will be found on 
pp. 171-183, Teachers’ Edition. 

For a still further description of the plan of 
the book please read the preface of each edition 


FIRST STEPS AMONG FIGURES. 5 


ADDITION TABLE, 5's and review. 


The following table is best learned by 
‘repeating each set of numbers many times, 
mame egeand 4 are 6, “2-and 4 are.6,’ etc., 
until it is well learned. ‘ Oft repeated, 
tong remembered.’ In reciting, it is best 
not only to say 2 and 4 are 6, but also 2 
from 6 leaves 4, and so on through the 
table, or at least until the pupil sees clearly 
that subtraction is the opposite of addition. 
If the table is copied.on the blackboard 
‘without the answers, the recitation may be 
‘conducted as in oral spelling, each pupil 
reciting a section, and any error being cor- 
rected in his turn by the first pupil who 
has noticed it. The teacher may ask the 
questions directly from the table in the 
book, if he prefers it. 

The letters are names for the sections, 
and are a convenience in assigning the 
lesson. 

When the pupils have learned the fol- 
dowing table, test their knowledge of it by 
the table in Teachers’ Edition, p. 63, which 
is differently arranged. 


6 FIRST STEPS AMONG FIGURES. 


a b Cc 
faa 3 ea 2... Same 
2 3 4 5 25S 4° Sufae 
Ome 579 3 aan 33 6-102 
a é op g 
YS ae | ee 4.25 $oae 
Se Ae 2 3 4 5 aes re 
7) ae © ee Cas 9 4 8 Zoe 


“2 and 4 are 6” may be written 2+4= 
6, and is read “2 plus 4 equals 6.” (Al- 
ways recite from below upward, that is 2: 
and 4, not 4 and 2). 

1. George has 3 cents and Mary has 5: 
cents; how many have both? 

Solution: They have the sum of 3 cents. 
and 5 cents, or 8 cents. 

2. John has 4 marbles and Henry has 
2; how many have both ? 

3. Susan had 5 pins and afterward 
found 4; how many had she then? 

4. A good boy brought in 4 armfuls of 
wood for his mother in the morning, and 3 
armfuls in the afternoon; how many arm- 
fuls did he bring in that day ? 


FIRST STEPS AMONG FIGURES. Th 


5. Nettie found 2 eggs in one nest and: 
5 in another; how many did she find in: 
both ? 

6. Helen had 3 sleigh-rides on Monday 
and 3 on Tuesday; how many did she 
have in the two days? 

7, Carrie had 1 needle and her mother 
gave her 5; how many had she then ? 

8. Charles rode down hill on his own 
sled 4 times and on his brother’s sled 4 
times; how many times did he ride down 
hill? 

Geeount by 2s.from 2 to 10, thus: 2, 
Bee O-6:..LQ. 

10. Read the following numbers, or 
write them in words and bring them to. 
Pie reeitation: 70; 94; 58; 85; 67; 93;-. 
Pepe O56 39; 155 88 59295 .13 5 303. 
Pige2) 5712534; 985 14. 

PiesCount by 2’s'from;2 to 20. 

Write in figures (Arabic): (12.) forty- 
six; (13.) fifty-three; (14.) eighty-one; 
(15.) sixty-nine; (16.) thirty-five; (17.) 
twenty-seven; (18.) fifty ; (19.) seventeen ; 
(20.) twelve The teacher should give 
more exercise of this kind. 

21. What cost an orange at 5 cents and: 
a lemon at 3 cents? 

22. Walter bought one book for 2 shil- 


See Teachers’ Edition, p. 64. 


my FIRST STEPS AMONG FIGURES. 


lings and another for 4 shillings; how 
much did he pay for both ? 

23. An inattentive pupil whispered 3 
times in the forenoon and 4 times in the 
afternoon; how many times did he whis- 
per during the day? 

24. On Nellie’s birthday her father and - 
‘mother each gave her five shillings; how 
much money did both give her ? 

25. Count by 2’s from 2 to 60. 

26. Write in figures (Arabic): VII; II; 
TV2 EAN Lieve les 

27. Write in letters (Roman): 5; 9; 
23 A527 3uLOy Oy sOhme le 

28. Count by 2’s from I to 9. 

29. Count by 2’s from I to 13. 

30. Add 2,.2, 1,2, 2,31, 2:52eee 
teacher will teach the pupil te write the 
numbers on his slate in a column, with a 
line underneath, and to write the answer 
‘beneath the line. 
31. Addtt2 to) 2a eo eee 
32. -Adds2 2 oie ie eee 
33: Odd tiyis geo oeee 


FIRST STEPS AMONG FIGURES. 9: 


SUBTRACTION TABLE.* 5's and review, 


“Oft repeated, long remembered.” 


a b Cc 
Peete 75 TO 9 
Brea.) 3 2 Sa a ol a a 
boo ae Ae 273 Bee Saved, 
d e ie a 
ee 7 Gea Coeetts O70 350 3 Ene. 
meee 5) 3 2 saa 
ee) 2 4A SB ly 4 2 


” 


“2 from 6 leaves 3” may be written: 
6—3=3, and is read ‘“‘6 minus 3 equals 3.” 

After this table is learned, test the class- 
by using the table in Teachers’ Edition, p. 
63, which is differently arranged. 

I. Count by 2’s from I to I9. 

2. Jesse having 7 cents spent 3 of them; 
how many had he left ? 

Solution: He had left the difference be-- 
tween 7 cents and 3 cents, or 4 cents. 


* The teacher will treat this table much as he did the- 
previous addition table. See the instructions with that” 
table. 


E me) FIRST STEPS AMONG FIGURES. 


3. Edward had 9g cents; he lost four of 
‘them; how many had he left ? 

4. Albert had 5 marbles, and gave John 
2 of them; how many had he left ? 

5. William has 10 cents and Lewis has 
5 cents; how many more has William than 
Lewis? 

6. Eight boys were skating on the ice, 
‘when three of them fell; how many re- 
mained standing ? 

7. Count by 2’s from I to 29. 

8. Read the following numbers: 100; 
163; 175; 158;.189; 107; 300ueeaas 
350; 416; 112; 761; 4608; QOGReagias 
070; 320; 510; 071; 60o: 

Write the following numbers in figures 
(Arabic): (9.) four hundred sixty-three ; 
(10.) two hundred thirty-nine; (11.) seven 
hundred eighty-one; (12.) one hundred 
sixteen; (13.) five hundred eight; (14.) 
-six hundred ninety; (15.) seven hundred 
twelve; (16.) three hundred twenty-one; 
(17.) five hundred; (18.) nine hundred 
‘two; (19.) five hundred sixty-one; (20.) 
-one hundred seven; (21.) the number of 
days ina year; (22.) eight hundred; (23.) 
one hundred nineteen; (24.) nine hundred 
forty. 

25. Count by 2’s from I to 69. 

See Teachers’ Edition, p. qo. 


FIRST STEPS AMONG FIGURES. II 


Write in figures (Arabic): (26.) XV; 
27 yy A XIT | en DOU 20s)tn KX LX * 
30.) XXIII; Pie e (22) UX (33.) 
XXIV: (34.) LV 

Write in letters (Roman): (35.) twelve; 
(36.) nineteen; (37.) fifteen; (38.) twen- 
ty-four; (39.) twenty-two; (40.) twenty- 
five; (41.) twenty-three. 

42. Count by 3’s from 3 to I5. 

Meeeeecaris: 2, 2,-2, 2, 2, 1,2) 2 

meee ie el) 212, 2. 1, 2,'2 

Sm)? 225515 92,) 2; 2,252 

emerges, 2. 7232 08, 1562; 2,2. 

Make and solve several examples like 
the preceding ones. 

47. A dwarf tree had 7 pears on it, but 
4 of them fell off; how many remained on 
the tree? 

48. Samuel’s mother gave him 8 cents 
to spend; he paid 4 cents for candy, and 
bought marbles with the remainder; how 
much did he spend for marbles ? 

49. Carlos started for school with 9 
marbles in his pocket, but when he got 
there he found he had lost all but 4 of 
them; how many had he lost ? 

50. Maggie had 6 needles; she broke 
3 of them; how many were unbroken? 

51. How many more wheels has a cars 
riage than a sulky? 


bd 


NN 


12 FIRST STEPS AMONG FIGURES. 


52. How many less fingers on one hand: 
than on both ? 

53. 8 is how many more than 3? 

54. Mr. Jones was idle 3 days of a. 
week; how many days of the week did he 
work ? 

55. Mr. Rawson worked 5 days one- 
week and 4 the next; how many days did. 
he work in the two weeks ? 

56. Julia misspelled 2 words in the fore- 
noon and 4 in the afternoon; how many 
did she misspell that day? 

57. Joseph’s mother told him he might. 
eat 2 apples; he ate 5 apples; how many - 
more did he eat than he ought? 

58. From Mr. Collins’s house to the- 
post-office it is 4 miles, and from the post- 
office to the school-house it is 3 miles far- 
ther; how far is it from Mr. Collins’s.. 
house to the school-house ’ 

59. Count by 3’s from 3 to 24. 

60. Add I, 2,.2;-2; 2,37 3c 

61. Add ‘2,2, 2, I, 2,2; 2)a2jenue enone 
rey 

62. Add'I, 2, 2, 1,'1, 2, 2yfe2ee2 ane 


63.. Add 5 2,:2, 2; 1} 2) 2:2) 3inete eee 
2. 
64. Add 2, 2,.2, 2, 1; 2, 2) 2ygpeeenieee 


See Teachers’ Eution, p. 66. 


FIRST STEPS AMONG FIGURES. 13 


65. Add I, 2, I, 2, 3, 3, 3, I, 2, 3, 3,2, 
1-2. °2. 

emcee 2 (2) yn 2) 2) 252) '2, 3,’ 3,2} 
ee. 

67. Susan took 3 peaches from a pile 
of 6 peaches; how many were left in the 
pile? 

68. A boy was sent to the store with 
8 cents; he spent all of them but 3; how 
many did he spend ? 

69. William spent 2 cents for nuts and 
5 cents for licorice; how much did he 
spend altogether? 

70. John hit James 4 times with snow- 
balls and James hit John twice; how 
many more times was James hit than 
John? 

71. Ina street car are 4 women and 5 
men; how many persons in the car? 

72. There were 2 pictures on one side 
of a wall and 3 on the other; how many 
on both? 

73. James had 8 doves; 2 pairs of them 
were killed; how many lived ? | 

74. A little girl has a cushion with 6 
pins on it; she took off 3 of them; how 
many were left ’ 

75. little boy earned 2 cents; he 
then had 7 cents; how many had he at 
first ? 


14 FIRST STEPS AMONG FIGURES. 


76. In a geography class, 4 pupils were 
in order and 3 were not in order; how 
many pupils in the class? 

When a number to be read contains 
more than three figures, place a comma 
before the third figure from the last; thus, 
3,216. It is read, three thousand two 
hundred sixteen. 

Read the following numbers: (1.) 5281; 
(2.) 6157; (3-) 9640; (4.) 7500; (5.) 
8609; (6.) 2050; (7.) 7008; (8.) 1000; 
(9.) 7318; (10.) 50713; (1I.) 4019; (12.) 
2800;  (13.) 5000; (14.) S8060;m=aae® 
7801}: (163): 31007. 

The teacher should show the pupil that 
in writing a number that contains thou- 
sands a comma should be placed after the 
number of thousands, and then the remain- 
der of the number should be written at 
the right, any of the three places which 
are not occupied being filled with ciphers. 

Write in Arabic: (17.) three thousand 
six hundred forty-five; (18.) seven thou- 
sand nine hundred fifty-one; (19.) one 
thousand two hundred seventy; (20.) 
eight thousand three hundred; (21.) two 
thousand twenty; (22.) six thousand one 
hundred five; (23.) nine thousand; (24.) 
five thousand seven; (25.) one thousand 


See Teachers’ Edition, p. 53. 


FIRST STEPS AMONG FIGURES. IS 


four hundred; (26.) seven thousand nine 
hundred four; (27.) five thousand forty ; 
(28.) eight thousand six; (29.) four thou- 
sand. 

30. Count by 3’s from 3 to 60. 

Write in Arabic: (31.) XXXVI; (32.) 
Bettas.) LV ; (34.) XL; (35.) LXII; 
peers NV ITs 637.) -L XIX; - (38.) 
ioeeney Ips 130.) XIX. 

Write in Roman: (40.) twenty-eight; 
(41.) fifty-four; (42.) forty-eight; (43.) 
seventy-seven; (44.) eighteen; (45.) four- 
teen; (46.) eighty-six; (47.) thirty-two; 
(48.) sixty-nine; (49.) fifty-seven; (50.) 
sixteen; (51.) thirty-four; (52.) seventy- 
five; (53.) forty-seven; (54.) eighty- 
eight; (55.) sixty-nine. 

56. Count by 3’s from I to 61. 

57. Count by 2’s from 2 to 60. 

58. Count by 2’s from I to 61. 

59. Count by 3’s from 3 to 60. 

The teacher should give a great deal of 
practice in counting as in the four exam- 
ples above. 

The teacher will show the pupil that the 
numbers in the following examples are to 
be so written in columns that the right 
hand figures shall be in one column. Add 
the right hand column first. Write below 
it the right hand figure of the result and 


16 FIRST STEPS AMONG FIGURES. 


add the left hand figure to the next col- 
umn. | 
60. Add 31, 23, 30, 2, 33, 14neemaiumae 

61. Add 22, 31, 12, 23, 3).20neqaumem 
and 23. 

62. Add 33, 20, 2, 13, 33))30;0eemeae 
and 30. : 

63. Add 23, I, 33, 20, 21,,;24l0—aasuae 
and 33. 

64. Add:13, 20, 23, 2, 20,1273 yeg—eam 
and I. 

65. Add 33,30, 2, 23, 20, 33, Sy aaamman 
29D angie. 

66. Add 32, 13, 33; 20; 12, 32zummeem 
21,° 13, 325anGuss 

67, Add 30, 2, 33, 21, 32, 23, 3, 30, 33, 
2133 Sanda. 

68. “Add 23,33, 20, 12, 3, 32;g0scmmnae 
32, 33, and 23. 

1. In a certain street there were 10 
houses, but last night 5 of them were 
burned; how many remain ? 

2. Fannie had 5 pinks; she gave three 
of them to Jane; how many did she keep ? 

3. A boy could not tell how many 5 
and 3 and 2 are. Can you? Show that 
you can. | 

4. Kate had adime; she spent Io cents 
for a doll; how many cents had she left » 

5. There were 7 boys on a bench, 5 of 


FIRST STEPS AMONG FIGURES. 17 


whom were studying; how many were 
idle? 

6. Fred had 5 cents left after spending 
4 cents; how many had he at first ? 

7. Four boys and three girls brought 
their lunch to school; how many do you 
think stayed at noon? 

8. Sherman had 3 books given him on 
Christmas; he had 3 before; how many 
has he now? 

g. Nine boys were playing soldier; 5 
of them were called home; how many 
were left to play? 

10. 8 apples are 4 more than James has; 
how many has he? 

11. Harry jumped 3 feet and Jimmy 
jumped 2 feet farther ; how far did Jimmy 
jump? 

12. A bad boy threw 5 kittens into the 
pond; 3 of them swam to the shore; how 
many were drowned ? 

13. Six boys were snow-balling; two 
of them were hurt so they would not play 
any more; how many finished the game? 

14. Bennie is 5 years old and Carrie is 
4 years older; how old is she? 

Peeed31, 23, 21, 3,33, 20, 12, 32, 
23, 3, and 33. 

Momence hs? 3, 21, 33, 30, 23, 12, 22, 
20,42, and. 3, 

See Teachers’ Edition, p. 67. 


18 FIRST STEPS AMONG FIGURES. 


W7, Add 3, 32,.°13; 33,.20)) yaaa 
227721033 and ale, : 

13. Add 31, 23, 33,12, 137 22,sqqeeee 
237 at Be eanies ss 

19. Add 2, 23, 33, 20, 13,920; 3musae 
2ON2 Tp 14/2 a candae ty 

20, Add 32, 13,-22, 33,3) 2iysommaae 
TR ene AN ces ee 
1. Count by 3’s from 2 to 62. 
2. Count by 4’s from 4 to 20. 


$4 
fs 


MULTIPLICATION TABLE. 5's and review. 


‘“ Oft repeated, long remembered,” 


a 1 c 
4 2 5 Ay eet 2 25°95 
2 3 4 Dies Acts 4 5 2 
S$). OY 20. eee re 8 250 4m 
a é 
2 At 5 eae 
5 64 2 3 4 
LO-5t Oe 3 100 Ouest 
af ete Mat 
41a 2 ee 
53 SO RNS 4 5 
AOE Ee 8 BL [2 356 


FIRST STEPS AMONG FIGURES. 19 


“2 times 4 are 8” may be written 4X 2= 
8, and is read “4 multiplied by 2 is 8,” 
or ‘‘2 times 4 equals 8.” 

At least some of the recitation of the 
above table should be as follows: “2 times 
wea eee in 6 tour times; “3: times 2 
are 6, 3 in 6 twice,’ etc. Always recite 
these tables upward, that is, 2 times 4, 3 
times 2:-4/times's, etc. 

After learning this table test the pupils 
by using the table in Teachers’ Edition, p. 
63, which is differently arranged. 

I. John bought 5 pencils at 4 cents each; 
what did they cost? 

Solution: If one pencil costs 4 cents, 
five pencils will cost 5 times 4 cents, or 20 
cents. If preferred, take the following: 
They will cost 5 times 4 cents, or 20 
cents, 

2. What cost 3 oranges at five cents 
each ? | 

3. If one sheet of paper costs 2 cents, 
what will 4 sheets cost ? 

4. A boy bought 3 marbles worth 2 
cents each; what should he pay for them ? 

5. How many pints in 5 quarts ? 

Show that there are 2 pints in a quart 
by pouring 2 pints of water, one at a time, 
into a quart cup. Show ina similar man- 
-ner that there are four quarts in a gallon. 


See Teachers’ Edition, p. 71. 


20 FIRST STEPS AMONG FIGURES. 


Solution: In one quart there are 2 pints, 
in five quarts there are 5 times 2 pints, or 
IO pints. 

6. How many quarts in 3 gallons? 

7, A lady bought 3 quarts of milk; she 
has only pint tickets; how many. tickets 
should she give for the milk ? 

8. If a pig should be fed 3 ears of corn 
at one feeding, how many ought he to have 
at 5 feedings? 

9g. Harry has 5 cents for picking one 
bushel of hops; how many cents will he 
have for picking 3 bushels? 

10. A boy’s mother gave him 3 cents 
for each armful of wood he brought in; he 
brought in 4; how many cents did he 
earn? | 

11. Three pupils have each 3 books; 
how many books have they altogether ? 

12. Jane has two dolls; Mary has 3 
times as many; how many has Mary? 

Write in Arabic: (13.);CXT3"( 1a jecee 
{15.) CL; (16.) CXXV; (17, VG Nees 
CXIX; (19.) CLXXIV ; 420.) EXOGen 
(21.) CLV 3'(22.) XLVI] "(226) Gi eee 
CUXT: 

Write in Roman: (25.) ninety-six; (26.) 
one hundred seventy-six; (27.) one hun- 
dred eight; (28.) ninety-nine; (29.) one 
hundred forty-eight; (30.) one hundred 
eighty-two. 


FIRST STEPS AMONG FIGURES. 21 


Read the following numbers: (31.) 
25758; (32.) 740515; (33.) 16300; (34.) 
$1407 ; (35.) 40520; (36.) 60000; (37.) 
70030; (38.) 9010; (39.) 18002; (40.) 
27643. 

Write in Arabic: 

- 41. Fifty thousand two hundred nine- 
teen. 

42. Twenty-seven thousand thirty. 

43. One thousand one hundred. 

44. Sixty-five thousand four hundred 
minety. 

45. Seventy thousand twenty. 

46. Forty-six thousand ninety-ovze. 

47. Nine thousand eight. 

For more practice see Teachers’ Edition, 
page 69. . 

48. Count by 4’s from 4 to 60. 

49. Count by 4’s from I to 17. 

50. Add 23, 33, 20, 3, 22, 13, 2, 33, 21, 
and 23. 

Si. Add 2,:33, 23, 31, 12, 3, 30, 23, 12, 
aman 32. 

52. Add 33, 21,13, 23, 3, 32, 23, 1, 23, 
30, 23, and 33. 

53. Add 23, 33, 30, 22, 13, 23, 30, 13, 
2.8 5, and-20, 

54. Add 3, 31, 23, 2, 33, 30, 23, 32, 22, 
32, 13, and 33. 

See Teachers’ Edition, pp. 60, 61. 


22 FIRST STEPS AMONG FIGURES. 


55. Add 32, 21, 33,3) 32; 20.3 .aeaam 
i 3)022, 3,-and:2! 

56. Add 2, 33, 21, 32, 13, 3, 23, 12, 32, 
B32 8 O anes 

57- Add 31, 12, 23, 33, 2323) e3 (gues 
23522) Ji fan ss 

1. In a school-room there are 5 desks 
in each row, and 4 rows; how many desks. 
in the room ? 

2. Lester wasted 5 minutes in school 
every day; how much tiie did he waste 
in a week? 

3. Three families of mice live in the 
garret; there are five mice in each family ;. 
how many mice in the garret? 

4. How many quarts in 5 gallons? 

5. If John picks 5 quarts of cherries one 
day and 4 quarts the next, how many does. 
he pick in the two days? 

6. A mother who had 5 hungry boys. 
made 2 loaves of bread one day, 3 the next, 
and 4 the third day; how many loaves did 
she make in the 3 days? 

7. There are 3 boats on the lake, and 4 
boys in each boat; how many boys on the 
lake ? 

8. What cost 3 balls, if one ball cost 3 
shillings ? 

g. James had 7 buttons on his coat; he 
lost off 3; how many remained on? 


FIRST STEPS AMONG FIGURES. 23 


10. There were 8 panes of glass in a 
window; a boy broke 2 of them with his 
ball; how many whole ones were there 
then ? 

11. A man paid 5 dollars for 2 dogs; 
one of them cost 3 dollars; what did the 
other cost ? 

12. Katie bought 3 yards of ribbon at 
2 cents a yard to trim her doll’s bonnet; 
how much did the trimming cost ? 

13. If one top costs 4 cents what will 
5 tops cost? 

14. I exchanged a cord of wood worth 
7 dollars for a ton of coal worth five dol- 
lars; how much did I lose? 

15. Two boys went nutting; one 
brought home 5 pecks and the other 4 
pecks; how many pecks did both bring? 

16. Frank paid 4 shillings for a pair of 
doves and 2 shillings for oats to feed them ; 
how much money did he spend for them? 
_.17. Emma, Hattie, and Lucy have each 
4 dolls; how many dolls have they all? 

18. Count by 4’s from I to 61. 

19. What will it cost to ride 4 miles on 
the cars, the fare being 2 cents a mile? 

20. How many times will the hands of 
a clock go from XII to VI in 4 days? 


24 FIRST STEPS AMONG FIGURES. 


DIVISION TABLE. 5's and review, 


“ Oft repeated, long remembered.” 


To be recited “5 in 25 five times,’ “2 
in 6 three times,” etc. 


c 
PA AGRE GTS) lO 3a 0 “Sire 
eae 4 3 2 3 4neee 
Sie bees eet 4 1 5 3 2 4 
ad 
Zhi ee 6 4 
2 hy Bre. Se a 
jee cy 2 SA ed 
I 5 
20ers 12s 
2 mae 4 5 
ree a a 342 


“s in 25 five times’ may be written 
25+5=5, and is read ‘‘25 divided by 5 
equals 5.” 

1. At 3 cents each how many lemons 
can you buy for 12 cents? 


See Teachers Edition. pp. 63, 73. 


FIRST STEPS AMONG FIGURES. 25 


oe. 


Solution: If oze lemon costs 3 cents, 
for I2 cents you can buy as many lemons 
as 3 is contained times in 12, or 4. If 
preferred, use the following: As many as 
piereedres3 S if). 12;-or 4; 

2. 10 cents will buy how many marbles 
at 2 cents each? 

3. Joseph spent 12 cents for oranges, 
paying 4 cents for each orange; how 
many did he buy ? 

4. A boy sold a pair of rabbits for 25 
cents; how many oranges at 5 cents each 
can he buy with the money? 

5. Mr. Brown paid a boy 12 shillings 
for work, at the rate of 2 shillings a day; 
how many days had the boy worked ? 

6. Lottie spent 16 cents for candy; she 
gave 4 cents an ounce; how many ounces 
did she buy? 

7. At 3 cents each how many marbles 
can Edward buy for 15 cents? 

8. Mary’s brothers gave her 16 cents, 
each giving her 4 cents; how many broth- 
ers had she? 

g. Harry has 8 dollars in the bank; his 
father has put 2 dollars there for him each 
birthday; how many birthdays has he 
seen P 

10. If one pineapple costs 2 shillings, 
how many can you buy for Io shillings ? 


26 FIRSI STEPS AMONG FIGURES. 


11. How many ink wells at 3 cents each 
can you buy for 9 cents? 

12. If the fare on the cars is 4 cents to 
a certain village, how much is the fare for 
both ways? 

13. Eight little girls were in the woods 
looking for violets; only 3 girls found 
any; how many found none? 

14. How many quarts in 3 gallons? 

15. How many pailfuls of beans will it 
take to fill an eight-quart basket, if each 
pail holds 2 quarts? 

16. Fred has 5 apples, John has one, 
and Harry has 3; how many have all? 

Read the following numbers: (17.) 
321468; (18:).. 108320; *(10:) s71Gs ame 
{20.) 400750; (21.) 604025 ; (22.) 700006 ; 
(23.) 800000; (24.) 70016; (25.) 215000; 
(26.‘ 380500; (27.) 50000. 

It is an excellent exercise for the class 
to write the foregoing numbers in words. 

Write in Arabic: 

28. Forty-nine thousand seven hundred 
sixty. 

29. Ten thousand ninety. 

30. Three hundred seventeen thousand 
nine hundred thirty-one. 

31. Nine hundred thousand one hun- 
dred one. 

32. Four hundred thousand forty. 


FIRST STEPS AMONG FIGURES. 27 


33. Six hundred thousand. 

34. Iwo hundred ninety-one thousand 
fives <- 

35 Thirty thousand ten. 

peed 32,21, (13, 3, 32, 22, 12,-33; 
ee a.023, and. 2. 

B90 E735 31;.23;.3, 33, 12; 2, 33) 21, 
a2. 33, and 23. 

Bue od 33,21, 23; 31, 12, 3, 23,13, 
e270, 3 2,,and J2. 

Pomoc 1, 23,°3, 12, 30, °23, 33,21, 
13, 33, 21, and 33. 

40. Add 32, 3, 23, 33, 13, 21, 23, 33; 
Bon20, 12) 2, and 21. 

eee 23,33). 2; 30,733, 13, 11, 32, 
Pee et.and 32. 

pee edditA3, 21, 4,714, 32,23, 42, 34, 
24, 42, 33, and 4. 

43., Add 4, 23, 42) 21,3, 44, 43, 23, 31, 
Pf and 31. 

I. What cost 3 books at 4 shillings 
each ? 

2. Mary rode in the swing five times, 
and Jane 4 times; how many times did 
they both ride? 

3. How many marbles at two cents each 
can you buy for 6 cents? 

4. Jane saw five doves on the ground; 
three of them flew away; how many re- 
- smained on the ground? 

See Teachers’ Edition, p. 69. 


28 FIRST STEPS AMONG FIGURES. 


5. Bessie went to school 5 days, and 
Mary went 3; how many more days did 
Bessie go than Mary? 

6. How many times can I take 2 mar- 
bles from a pile of 8 marbles? 

7. A little girl had 20 cents; how many 
four-cent lead pencils can she buy with her 
money? 

8. A boy walked 2 miles each day for 
4 days; how far did he walk? 

g. A boy having three five-cent pieces. 
lost two of them; how many cents had he 
left ? 

10. A little girl ate 3 buckwheat cakes 
for breakfast, and her brother ate 4; how 
many did both eat? 

11. Lewis gave 4 boys 4 marbles apiece ;. 
how many did he give them all? 

12. Jennie spent 5 cents for raisins, 3 
cents for candy, and 2 cents for a stick of 
gum}; how much did she spend ? 

13. On the east side of a house there 
are 5 windows; 3 of them are open; how 
many are closed ? 

14. A good boy carried 8 pails of water 
for his mother on Monday, and 5 on Tues- 
day; how many more did he carry on 
Monday than on Tuesday ? 

15. How many quarts in 4 gallons? 

16. Among how many children can Ff 


FIRST STEPS AMONG FIGURES. 29 


divide 15 plums that each may receive 3 
plums? 

17. There are 16 towns in Ontario 
County; if you learn the names of 4 of 
them each day, in how many days will 
you learn the names of all of them ? | 

18. Charles had 4 marbles and his broth- 
er gave him 5; how many had he then? 

19. Henry had to stay after school 5 
minutes for whispering, and 2 minutes to 
solve an example; how long did he have 
to stay ? 

20. If John gets 5 cents for husking one 
bushel of corn, how many bushels must he 
husk to earn 20 cents? 

21. If a boy traded a knife worth Io 
cents for a top worth 5 cents, how much 
did he lose? 

22. How much will a boy earn in 4 
days at 3 shillings a day? 


ADDITION TABLE. 7's and review. 
a b 

Ome 5 2.42 O° 3 

gipsdas 5: 0 423 455 

Geert? 1! On 7 iO to 


See Teachers’ Edition, p. 74. 


30 FIRST STEPS AMONG FIGURES. 


"5 d 
Feet, rptaoks 
te 17 (33 ee 3 Fees 
as bp eee me LO 4h 

e je 
S224 CAG eS 
SEATS ab he kO Fi 
$5 20g. Oneibe 1O-tag 

g ht 
swey) 7 2 
VER Naas 5 6 
13 Oct TOS 


a 
Ge tana ese it 8! IO 
Oo Jig Oat 3 
Te iO aed, rf 
c 
LA-20% 7omi2 II 
7: Sp age 6 


i) 


FIRST STEPS AMONG FIGURES. 31 


e op 
See Ow 10.13 Ostons OF.-LO 
Ba i Z Sie 40% 35 
“Gs eee Syce tra Zee § 

g h 

i SNe imate) Gine on Lt 

Moe Onie7 

O45 Oe 26a 4 


MULTIPLICATION TABLE. 7’s and review. 


a b 
fest 7S Apes Gas On 3 
Sede SO Ve ewer Martie 
eee 35.. 30 Ape 2 22 4S 

C a 
eee 4 7h EMS eens ery: 
OD es ae Ub EAC ES ili) 
42 35 6 16 Zit 0 315) 740 

€ ap 
m2. 4 «=O se a EPIC 
teete 5 0 Tan As vote 


ooo 20; 30 2INs2O ee LOe eed, 


32 


DIVISION TABLE. 


FIRST STEPS AMONG FIGURES. 


S 
O SKS ray 
fk See 
AZO se 28 


a 
Bop Backed | 
OSes dine 
B55 On eZ. 
Cc 
Su 20: sO 
rie ep Xs. 
2a ASO 
é 
Qh Camel 
ene Pe aS. 
7: i503 ae 
& 
A2 107028 
Y etek ear! 
ODES en, 


h 

5 2 
5 6 
PANE A: 


b 
16-6 <35 
4. TS ie 
2 ‘we 
a 
152° 24> 8 
54 Se 
3: © Gree 
J: 
IO’ 200827 
5 Ae 
2. See 
h 
25. T2ueeas 
5 Omaae 
5) aes 


See Teachers’ Edition, p. 95. 


FIRST STEPS AMONG FIGURES. 33 


To the teacher. Read carefully the pre- 
faces to both the Pupils’ and the Teach- | 
ers’ Editions, and also the Special Notice 
which precedes the latter. 
Solve the following examples in ad- 
dition :-— 
Meee 03 314 31) 5, 32. 6-21 
31 23 33 13 13 13 
55 31 22 12 e3 32 


20 20 30 33 21 33 
13 13 23 30 12 20 
32 23 Ia 22 32 32 
23 32 32 31 23 3 


31 23 31 13 13 31 
33 31 23 23 32 31 
12 33 33 31 23 23 


7iss2 8. 33 O37) 3107 32 


23 31 12 23 
3! 23 33 31 
33 33 22 23 
21 12 31 13 
13 23 23 31 
22 ral 12 20 
33 32 od 33 
31 23 32 23 
23 33 23 31 


——— — et — 


Some of these examples are to be given 
daily. See Teachers” Edition, p- 88. 


34 


FIRST STEPS AMONG FIGURES. 


Solve the following examples in ad- 
dition :-— 


11.232 
21 

13 

Bo 

30 

2 

23 

31 

23 

ee 


S33 


23 


16. 


PL RFS 

21 

32 

35 

13 

32 

3 

12 

30 

21 

33 

32 

Suet: 
23 
23 
30 
23 
31 
22 
32 
21 
32 
33 


Ighize 
By: 
31 
22 
33 
21 
13 
2I 
Be 
Be 
13 
33 

2 eGert oO: 

22 

30 

21 

13 

Los 

33 

22 

31 

23 

22 

31 


14.63 39000 fens 
22 23 
31 30 
23 33 
20 22 
32 31 
31 ra 
23 33 
33 20: 
31 3 
23 31 
oa 33: 
217510. 
33 23 
32 31 
23 23° 
33 30 
ze 33 
aa 21 
23 13 
32 32 
13 32 
33 23 
31 32 


Some of these examples are to be given: 


daily. 


See: Teachers’ Edition, p. 89. 


FIRST STEPS AMONG FIGURES, 


35 


Solve the following examples in :id- 
dition :— 
20232 


13 
23 
31 
33 
23 
31 
21 
12 
32 
33 
32 


24. 33 


21 
30 
13 
23 
3I 
30 
23 
31 
LZ 
23 
32 


21. 


25. 


33 
21 
33 
32 
22 
23 
ei 
32 
13 
at 
23 
35 
23 
BI 
22 
30 
3 
32 
52 
23 
33 
31 
32 
33 


22. 


206. 


31 
23 
a3 
31 
22 
a2 
23 
21 
32 
33 
32 


aye 


al 
23 
32 
21 
13 
33 
Se 
20 
13 
32 
33 
23 


ope 
31 
23 
oe 
30 
23 
12 
33 
23 
32 
om 
23 
27.513 
53 
21 
a2 
33 
23 
33 
20 
12 
33 
32 
12 


Some of these examples are to be given 


daily. 


36 FIRST STEPS AMONG FIGURES, 


Solve the following examples in ad- 
dition :— 


28. 32 20.) 32 30. 21 213% 


20 2% 32 23 
33 32 23 30 
23 33 33 22 
I2 21 Pes | i Ye 
33 13 220 33 
31 33 33 21 
Pes 20 13 21 
33 22 23 13 
32 31 33 33 
21 33 21 30 
33 23 32 22 
22.230 3321 34s 
23 33 32 
12 12 13 
31 23 23 
33 32 32 
22 31 33 
32 32 23 
23 20 32 
31 II 30 
33 32 31 
22 23 23 
31 33 22 


Some of these examples are to be given 
daily. 


FIRST STEPS AMONG FIGURES. OL 


After these examples have all been solved 
once, unless the pupils are very ready in 
adding, let them commence at the first 
example and solve them all again. It is of 
the utmost importance that the pupil should 
learn to add quickly and correctly, for in 
practical life he will use addition a dozen 
times where he will use fractions once. 


*Examples in Subtraction. 


35. 75986 36. 96897 37. 68795 
54234 72044 5250 


38. 69587 = 39. 95786 ~—s go. 68795 
47043 2053 6203 


41. 86975 42. 79586 43. 68579 
20324 2102 40234 


a 


44. 76859 45.97586 46. 68579 
4506 35430 5345 


*Examples in Multiplication. 
ie 92 130,23 F 3.°13,202 4. 21,032 
2 3 3 2 


See Teachers’ Edition, p. 9I. 


38 FIRST STEPS AMONG FIGURES. 


So 31,023 6.0n tence 20,123 3. 2RPS2 
3 2 6 3: 


9 31,420 10. 32,301 If. 30,423 12. 23,123. 
2 3 2 3. 


*Examples in Division. 


13. 2)64.682 14. 3)93,609 I5. 2)20,486- 


16 2)40,826 17. 3)39,069 


18. 2)82,604 19. 2)28,064 


20. 3)90,639 21. 2)26,048 


22. 2\80,264 23. 3)60,936 


See Teachers’ Edition, p. 92. 


FIRST STEPS AMONG FIGURES. 3 


= 


*Examples in Addition. 
24. 342 25. 234 . 260. 434 


231 423 244 
423 344 342 
443 244 444 
334 432 322 
234 343 434 


422 443 324 


275,213 28. 444 29. 431 


432 231 242 
444 443 344 
344 232 423 
231 424 344 
414 344 332 
342 231 Agios 


os ——— —— 


Some 31. 2439 320-4 


442 424 342 
321 331 234 
434 212 23 
444 443 341 
343 344 4.2 
212 424 444 
243 341 341 


444 234 213 


* The teacher is to show the pupils how to solve such 


exarr ples. 


4o 


G7 246 


FIRST STEPS AMONG FIGURES. 


33. 


36. 


134 
423 
344 
103 
422 
344 
441 
343 
413 


2342 
4324 
3432 
2343 
3423 
2343 
3434 
4232 
1443 
3234 
4342 


See Teachers’ dition, p. 93. 


34. 


37- 


324 
431 
344 
222 
431 
444 
324 
25 
444 


4323 
2433 
4344 
4224 
3433 
1234 
4322 
2443 
3244 
4432 
2343 


35. 


38. 


432 
341 
2235 
212 
434 
244 
443 
332 
423 


3 eee 
eee 


At 


FIRST STEPS AMONG FIGURES. 


39. 


3424 40 4234 
4132 3442 
3434 2323 
1234 4342 
4343 1234 
2432 3421 
1243 4343 
3324 2422 
4432 3234 
3243 4343 
4432 2342 

AzyBAUE 43.431 

234 343 

443 234 

444 321 

321 443 

213 444 

442 324 

344 242 

431 441 

342 304 

214 323 

404 444 

321 


4B 


42 


FIRST STEFS AMONG FIGURES. 


A421 5. 4s act 40. 423 
441 232 244 
334 444 231 
234 323 444 
443 213 324 
342 441 431 
134 344 344 
423 422 444 
444 343 321 
324 243 423 
443 412 344 

22 333 442 

203 431 322 
47. 3424 48. 4234 49. 4324 
4323 3342 3432 
32.43 4433 4243 
4434 1234 2324 
2332 3243 4433 
3443 4324 3442 
4324 2442 2334 
2423 3334 4343 
4244 4234 2434 
3423 2442 2343 
3342 3223 4123 
4324 4324 3434 
3423 3443 4442 
2342 4321 3223 


See Teachers’ Edition, p. 94. 


FIRST STEPS AMONG FIGURES. 43 


50. 3423 51. 3443 
3244 4234 
4334 2343 
2433 4324 
4242 4432 
3423 2444 
4344 4321 
1234 3443 
4422 4242 
4342 4433 
3423 3444 
4444 2344 
3444 4323 


2331 3434 


I. Richard has 7 cents and Oliver has 
6 cents, how many have both ? 

2 Cora bought 5 sticks of candy and 
Hattie 4, how many did both buy ? 

3. Herman had 11 cents, but he lost 3 
of them, how many had he then? 

4. Herbert had 7 marbles, he found 5 
more, how many had he then? 

5. Ella had 8 new needles, she broke 5 
of them, how many whole ones had she 
then ? 

6. Clara solved 12 examples while Kitty 
solved 7, how many more did Clara solve 
than Kitty? 

See Teachers’ Edition, p. 97. 


44 FIRST STEPS AMONG FIGURES. 


7. Anna had 10 oranges, she gave away 
4 of them, how many had she left? 

8. Frank had 4 pencils, he bought 3 
more ; how many had he then? 

9. What cost 7 lemons at 4 cents each ? 

10. If an orange cost 5 cents, what will 3 
oranges cost ? 

11. If an orange cost 6 cents, and a 
lemon cost 5 cents, what will both cost ? 

12. How many quarts in 3 gallons ? 

13, Henry had 13 apples, he gave his 
brother 7 of them ; how many had hethen ? 

14. At 5 shillings a bushel, what will 6 
bushels of apples cost ? 

15. What cost 7 pencils at 3 cents each? 

16. A boy walked 4 miles in the morn- 
ing and 3 miles in the afternoon. How far 
did he walk ? 

17. Samuel has 7 cents in one pocket 
and 5 cents in another, how many in both ? 

18. Carlos had 9 cents in his pocket, he 
lost 4 of them through a hole in his pocket, 
how many had he then? 

19 Mary spelled 8 words correctly and 
Emma spelled 6 words correctly, how many 
more did Mary spell correctly than Emma? 

20. At 3 cents each, how many marbles. 
can you buy for 15 cents? 

21. 12 boys were sliding on the ice, 7 of 
them fell, how many remained standing? 

See Teachers’ Edition, p. 98. 


FIRST STEPS AMONG FIGURES. 45 


22. If a boy earn 7 cents a day, how 
many days will it take him to earn 42 cents ? 

23. If one lead-pencil cost 6 cents, how 
many can be bought for 30 cents? 

24. James had 12 cents, he spent 5 of 
them and lost 3 more, how many had he 
left ? 

Solution: He disposed of the sum of 5 
cents and 3 cents or 8 cents. He had left 
the difference between 8 cents and 12 cents 
or 4 cents; or, ifhe spent 5 cents and lost 
3, he disposed of the sum of 5 cents and 
3 cents or 8 cents. If he had 12 cents and 
disposed of 8 cents he had left the difference 
between 12 cents and 8 cents or 4 cents. 

25. Sarah had 10 needles; she gave 3 
of them to Nellie and 4 of them to Martha. 
How many had Sarah left ? 

26 Herbert had 6 cents, he earned 7 
cents and spent 5 cents; how many had he 
then? 

27. Joel had 7 cents, he earned 5 cents 
and found 4 cents ; how many had he then ? 

28. How many days will it take Walter 
to walk 28 miles, if he walks 4 mileseeach 
day ? 

29. An orange cost 6 cents and a cocoa- 
nut 3 times asmany. How much did the 
cocoanut cost? 

2* 


46 f1RST STEPS AMONG FIGURES. 


30. Mary had 11 cents, she lost 5, and 
earned enough to make her number 9. 
How many did she earn ? 

31. There are 4 columns in John’s spell- 
ing lesson and 5 words in each column. 
How many words in his lesson ? 

32. Some boys are out flying kites, the 
wind blows down two kites and 7 less 3 re- 
main. How many at first in the air? 

33. A boy hoed corn for 4 cents a row 
and earned 24 cents. How many rows 
did he hoe? 

34. A mancan walk amile in lominutes, 
he starts from his home and walks to town 
in 5 minutes. How far from town does he 
live ? 

35. [had 1ocents. I bought 2 two-cent 
stamps, and gave 4 cents to a poor little 
boy for bread ; how many left for candy ? 

36. There are g ten o'clock scholars this 
morning and one more than one third of 
them left their books athome. How many 
of them brought their books ? 

37. Willie had 13 cents and he spent 6 
of them for candy ; how many had he left? 

38. How many sponges at 6 cents each 
may be bought for 36 cents? - 

39. What cost a pencil at 7 cents and a 
pint of peanuts at 5 cents? 


FIRST STEPS AMONG FIGURES. 


47 


40. What cost 7 pencils at 5 cents each? 


Count by 5’s from 2 to 62. 


Review counting by 5’s from I and 5 to 


61 and 60. 


1. 345 
452 
553 


Review counting by 4’s. 
Count by 5’s from 3 to 63. 
Count by 5’s from 4 to 64. 
4 


553 


a5 


454 


542 
135 
523 
454 
542 

5 
352 


534 — 


435 


_—— 


- 505 


453 
344 
Ei 
413 
aia 
435 
214 
543 
345 


3. 405 
352 
544 
255 
533 
441 
355 
523 
344 
555 
434 


6. 305 | 


543 
444 
354 
. 532 
445 
354 
543 
455 


544 


353 


——_—— 


48 


FIRST STEPS AMONG FIGURES. 


Examples in Subtraction. 


ifs 


II. 


13. 


15. 764,358 
201, 


79,687 
64,252 


- 79,689 


43,204 


795,869 
43,543 


96,758 
54.428 


223 


18. 


20. 


2222 O12 
46, 


647,094 
24,070 


74,087 
30,435 


566 


16. 764,037 


23,010 


613,021 
56,554 


23. 


8. 96,487 
31,343 


10. 687,985 
34,532 


12. 47,685 
5,232 


14. 5,879 
2,343 


17. 423,703 
12,40C 


19. 74,087 
3,053 


21. 695,047 
252,004 


24. 431,024 
55,357 


-_-——- ——_—_ 


See Teachers’ Edition, p. 101. 


rae Ay 


28 


34. 


37: 


40. 


B43. 


46. 


FIRST STEPS AMONG FIGURES. 


26. 


634,210 
56,643 


942,031 
66,406 


. 312,403 


45,257 


213,042 
36,527 


324,102 
46,637 


430,221 
52,147 


534,210 
16,436 


703,524 
20,352 


29. 


ao 


35. 


38. 


AT! 


44. 


47. 


820,132 
53,657 


731,203 
64,636 


420,314 
54,652 


314,253 
51,646 


231,430 
12,257 


320,413 
63,257 


130,241 
53,725 


423,102 
47,056 


27 


30. 


36. 


50: 


42. 


45. 


48. 


49 


431,201 
54,4604 


841,310 
63,653 


: 3415532 


34,354 


453,621 
32,365 


425,301 
51,625 


534,102 
62,637 


342,013 
30,157 


624,130 
OTs503e> 


50 FIRST STEPS AMONG FIGURES. 


If more practice is desired at this stage 
the foregoing examples may be reviewed. 
Count by 6’s from 6 to 60. 


I. 4532 2. 3541 3. 5423 
3215 4325 4352 
5453 5432 3544 
3344 3254 5435 
2535 5545 4334 
4253 3251 2453 
5432 4314 3545 
2354 2443 4334 
4235 5425 5442 
5542 4543 2355 

4. 3524 5. 4325 
4352 3552 
3445 5445 
5334 3453 
4523 4524 
5452 5335 
3445 3452 
4334 5344 
5523 2325 
4354 4553 


See Teachers’ Edition, p. 103. 


FIRST STEPS AMONG FIGURES. 


Ti 


6. 3542 
4354 
2435 
5043 
3530 

354 
‘4245 
5432 
3554 
4322 
2453 
5245 


245 
4532 
2454 
5321 
3543 
4254 
5435 
2353 

540 
3254 
4342 
5535 


Ome 5 442 
3534 
4253 
5425 

343 
4534 
2345 
5234 
4553 
3425 
5342 

2435 


ET 


10. 3452 
5544 
2°45 
4532 
5453 
1324 
5045 
3453 


51 


52 FIRST STEPS AMONG FIGURES. 


Multiply : 
TI2(2, 314 eee and 13.12,014 
2 2 2 

14. 30,124 * 15. 21,302: 

2 3 

16. 34,201 17. 23,041 

2 2 


The teacher should show the pupils how 
to solve the following examples: 


18. 35,246 19. 42,356 
2 3 


20. 46,352 21. 35,042 
2 3 


22. 36,452 23. 26,453 24. 46,352 
3 3 2 


25. 64,526 26. 53,624 27. 64,524 
3 4 3 


See Teachers’ Edition, p. 104. 


FIRST STEPS AMONG FIGURES. 53 

28. 53,426 29. 64,536 30. 35,642 
4 4 2 

31. 46,352 32. 36,426 33. 35,246 
3 4 3 
34. 36,452 35. 36,546 36. 64,526 
6 6 6 

37. 46,352 38. 63,524 39. 35,042 
5 4 4 

40. 40,536 41. 30,462 42. 46,035 
5 6 5 

43. 26,304 44. 25,030 45. 52,004 
6 6 6 


Count by 6’s from I to 61. 
Count by 6’s from 2 to 62. 
‘Count by 6’s from 3 to 63. 
‘Count by 6’s from 4 to 64. 
Count by 6’s from 5 to 65. 


54 FIRST STEPS AMONG FIGURES. 


ADDITION TABLE. 3’s and review. 


OlwNnN 
co; 
MINA g Olu og 


—_ 


mq | 


CO} Ww 
WwW} ON 
OV} CO ~ 
CO}W Un 


5 penne 
780 7 ae 
1341 1015 ago 


wir}; COON 
NY | O07 
WW |B Cols 
—" 

O};}uUIm 
OC} ON 
N 
ioe) 

Oo’ 


aH 
—_ 


SUBTRACTION TABLE. S’s and review. 
a 

LOM? Jal Omens 

55 43 OLO ae, 

6 


O21126,8.:10 Stas 
Onto es Oe 
Sa Go Bees 


Ss 
96.12.13 8. I0 10°97 . 12 sigameeie 
§3°:7. 54.3. 83 4. 35 Oee 
43 5 84° °7 263 9 Fea 
See Teachers’ Edition, p. 105. 


FIRST STEPS AMONG FIGURES. 55. 


- MULTIPLICATION TABLE. S’s and review 


a b C 
Wee 6307 4 8 5 2.6 2 
Smee 38 45 (8 7 8 3 
PAO eee O 14 A509 28 20 48. 35 16 18:12" 

a e ye 
fe eS. 2) 0°30 7.2 4, 8 5 2 
Bee 3A 6 6 7. 8 3 ANS 
ieee tee O4 158 30 18 40. 32 24 20 10 

g : h z 
eee 5 2-6 3 A 8 5 2 6 
pees 4 5. 027.8 (02057. 83.4 
SOrz 502 32 25 12 4224.24 5640624 
DIVISION TABLE. S's and review. 


a b C 
20°09 45 14:30 4016 21.20 48.35 160 18 12 
ee Oe 5 4 38 5 O87 83 
Bom e ee, 8 4) 7 48 5 2 0. 3 
ad 
6 2 


é 
35 24564215 24640 
Aes TO 

5 


2 4 
6 Sumo neo 
7 8 


5 
5 
8 5 


56 FIRST STEPS AMONG FIGURES. 


1. George paid 15 cents for a knife, and 
after breaking it sold it for 8 cents. How 
many cents did he lose? 

2. Lewis bought a reader for 6 shillings 
and an arithmetic for 4 shillings ; how much 
money did he spend for both ? 

3. Anna is 13 years old and Mary is 
5 years old. How many years older is 
Anna than Mary? 

4. John had 8 cents; he lost § of them, 
how many had he left ? 

5. What cost 4 books at 8 shillings each ? 

6. How many oranges at 5 cents each 
can you buy for 40 cents ? 

7, Walter spent 16 cents for pears at 2 
cents each ; how many pears did he get? 

8. Jane had 15 needles; she lost 3 of 
them and broke 5 ; how many had she left ? 

9g. Byron rode down hill 4 times one 
afternoon and his father twice as many 
times. How many times did Byron’s father 
ride down hill? 

10 William had 26 cents ; he lost two of 
them and spent the rest for marbles at 3 
cents cach. How -many marbles did he 
buy ? 

11. Charles was sent to the store to buy 
6 spools of thread at 8 cents a spool. He 
took 50 cents with him, how much change 


should he take home ? 
See Teachers’ Edition, p. 108, 


FIRST STEPS AMONG FIGURES. 57 


12. A boy having 8 cents earned 5 cents 
and his sister gave him 2 cents, how many 
had he then ? 

13. A boy set 2 traps. in the woods; 2 
rabbits: went into one trap and twice as 
many went-into another. How many went 
into both? 

14 A gun carriage has four wheels ;how 
many wheels have 7 gun carriages ? 

15. If it takes 6 horses to draw one can- 
non, how many horses will draw 8 cannons ? 

16 There are a sergeant and 6 privates 
at one picket post, and a corporal and 4 
privates at another. How many soldiers at 
both? 

17. If 3 oranges cost 12 cents what will 
one orange cost? 


Solution : If ¢ivee oranges cost 12 cents, 
one orange will cost one-third of 12 cents 
or 4 cents. 

We get 4 of a number by dividing it by 
3, 4 of a number by dividing it by 4, etc. 

What is one-fourth of 24? 

What is one-third of 21? 

What is one sixth of 42? One-fifth of 
30? One-fourth of 32? One-seventh of 
56?. One-half of 14? One-sixth of 18? 

18. If 8 horses eat 24 bushels of oats in 
a week, how many bushels will one horse 
eat? 


58 FIRST STEPS AMONG FIGURES. 


19. If 42 cents is the price of 7 marbles, 
‘what is the price of one marble? 

20. 4 boys have 28 cents, and each have 
an equal number. How many cents has 
each boy? 

21. A farmer has 18 pigs in 3 pens and 
the same number ineach pen. How many 
pigs in one of the pens? : 

22. If 8 pears cost 16 cents, what cost I 
pear ? 

23. At 4 cents each, how many lemons 
can be bought for 28 cents? 

24. If three’ tops cost 15 cents, what 
cost I top? 

25, WV Dateecost eat pieeel of oats, if 7 
bushels cost 35 shillings ? 

26. 12 dollars will buy how many birds 
at 2 dollars each ? 

27. If 5 hens cost 20 shillings what will 
1 hen cost? 

28. How many peaches at 2 cents each 
can you buy for 16 cents ? 


FIRST STEPS AMONG FIGURES. 


f. 5304 
4243 
3524 
5452 

SBE 
4343 
3425 
5254 

- 2542 
4335 
3454 
5543 
4325 


4 


See Teachers’ Edition p. 110. 


4305 
3450 
5234 
2543 
5455 
3213 
4543 
2355 
5432 
4343 
2524 
3455 
3532 


2. 


453 

304 
5035 
4540 
3235 
2424 
5353 
4545 
3424 
4353 
an32 
4245 
5454 


3. 5243 
4524 
3452 
5335 
3543 
4254 
5435 
3542 
5354 
4435 
3423 
5354 
3543 


—e 


5. 3544 
5434 
4355 
3243 
5424 
4552 
2345 
5433 
2253 
4534 
3445 
2343 
4554 


59 


60 FIRST STEPS AMONG FIGURES, 


6. 4305 7 2345 8.5435 
3453 5334 554 
435 4243 3443, 
5044 3545 5545 
3453 5434 4353. 
4545 3552 5534 
5432 3345. 3444 
3354 5453 5554 — 
3545 4245 2345 
34 3524 4235 
- 5432 4305 5453: 
4355 5353 3544 
4544 3445 5355. 
9. 3452 10. 3452 
2345 5435 
5443 4544 
4534 3353 
4455 5445 
5243 3234 
3524 5544 
4355 4353 
5533 2455 
4432 4434 
5454 5554 
3345 5432 


5432 3245 


——— 


FIRST STEPS AMONG FIGURES, 61 


Count by 6’s from 2, 3, 4 and 5, to 62, 


63, 64 and 65. 
Read the following numbers: 
I. 750000748 2. QOO000047 


3. 680000740 4. 700746000 
i 1 2007000 6. 750908716 
7. 801000071 8. 679374819 
9. 715016390 10. QOOO60000 
II. 7000000 12. 800000005 

Addition : 

Finding how many units there are in two 
or more numbers and expressing them in 
one number is called addition. 

The number found by addition is called 
the sum or amount... 

Proof. Add the columns both upward 
and downward, and if the results agree they 
are probably correct. 

11. Write in Arabic fifteen mil ten th. 
ninety. 

12. Write in Arabic three hun fitty mil. 
nine hun th. 

13. Write in Arabic eight mil. five hun. 
forty-three th. seven. 


14. Writein Arabic onehun mil six hun. 
thirty-two. 
15. Write in Arabic seventy-fye mil. 


three hun th. 


62 FIRST STEPS AMONG FIGURES. 


16. Write in Arabic fifty mil fifty th. fifty. 

17. Write in Arabic two hun. mil. sixty 
th. four. 

18. Write in Arabic one hun. five mil. 
one hun. five. 

19. Write in Arabic one hun. nineteen 
mil. forty th. 

20. Write in Arabic three hun. eight mil. 
thirteen th. two hun. eighty-one. — 

21. Write in Arabic twenty th. 

22. Write in Arabic fifty mil. fifty. 

23. Write in Arabic three th. two hun. 
forty-five. 

24. Writein Arabic nineteen mil.five hun, 
th. 3 
25. Write in Arabic eight mil. three 
hun. . 

26. Write in Arabic four hun. mil. 

27. Write in Arabic seven hun. sixty-one 
mil. five hun. sixteen th. twenty. 

28. Write in Arabic six hun. th. 

29. Write in Arabic fifty th. forty. 

30. Write in Arabic three hun. eight. 

31. Write in Arabic nine th. 

32. Write in Arabic six hun. mil. five. hun. 

33. Write in Arabic sixty mile, four hun. 
th. three hun. 

34. Write in Arabic three mil. fifteen th. 
thirty. 

See Teachers’ Edition, p. 114. 


FIRST STEPS AMONG FIGURES. 63 


a 


*35. Write in words 13,004,020. 

36. Write in words 300,216,000. 

37. Write in words 232,341,519. 

38. Write in Roman fourhun. sixty-three. 
39. Writein Roman eight hun. forty-four. 
40. Write in Arabic DCLXXVII. | 
41. Writein Romantwo hun. eighty-nine. 
42. Write in Arabic XCVIIL. 

43. Write in Arabic CDXI. 

44. Write in Roman seven hun. nineteen. 


Subtraction: — 
Taking one number from another num- 
‘ber is called subtraction. 


Remainder or difference. 


The number found by taking one num- 
ber from another is called the difference or 
remainder. 

In subtraction the number to be sub- 
tracted is called the subtrahend, and the 
number it is subtracted from is called the 
minuend. ‘The result in subtraction is 
called the difference or remainder. 

ProoF. Add the remainder and the sub- 
trahend, and if the result equals the min- 
uend the work is probably correct. 

The pupils should often be required to 


* Notice that such compound words as seventy-five, 
forty-one, sixty-nine, etc., require a hyphen. 


64 FIRST STEPS AMONG FIGURES. 


write the name of each number in examples 
in subtraction, as follows: 


Subtraction. 


(1) 
8,342,053 
474:377 


—— 


(4) 
71,420,035 
241,068 


(7.) 
74,200, 35 2 
5,402,568 


(10.) 


94,002,531 
7,006,763 


321 Minuend. 


45 Subtrahend. 


276 Remainder or dif. 


(2.) 
73,520,031 
243,075 


——, 


(5.) 
93,520,014 
5,700,043 


(8.) 
6,314,025 
565,068 


(AT) 
81,350,024 
576,546 


(3) 
81,400,253 
2,302,675, 


(6.) 
4,531,024. 
765,337 


(9.) 
73,500,241 
212,376: 


——s 


(12.) 
34,200,156 
7,620,478 


See Teachers’ Edition, p-. 118. 


FIRST STEPS AMONG FIGURES. 65 


(13 ) 
71:352,034 
181,376 


—=—— 


(16.) 


93,510,042 
46,350,357 


(19.) 
$3,001,425 
5,005,743 


(14.) 
95,300,421 
420,376 


(17.) 
35,200,416 
2,540,348 


(20 ) 
95,320,041 
4,460,474 


(22.) 


(15) 


863,005,241 


2,307,374 


(18.) 


71300,425 
602,557 


(21.) 
85,241,300 
71534,720 


94,300,52 I—570,376=? 


(23.) 


42,5 30,014—6,765,068=? 


(24 ) 


71,300,524— 1,000,677=? 


(25.) 


82,400,153—4,740,075=? 


66 FIRST STEPS AMONG FIGURES. 


The teacher will choose between the two 
following sets of definitions: 


Multiplication. 


A short method of adding equal numbers. | 
is called multiplication. 


Multiplicand. | 

One of the equal numbers is called the 
multiplicanda. 

Multiplier. 

The number which shows how many of 
the equal numbers are used in adding is. 
called the multiplier. | 

In the example 542+542+542, 542 is the 
multiplicand and 3 is the multiplier, and we 
solve the example as follows: 

542 Multiplicand, 
3 Multiplier, 


1,626 Product. 


Or the following definitions may be used: 
Multiplication. 


Taking a number a certain number of 
times is called mzultzplication. 


Multiplicand. 


The number taken, (or multiplied,) is 
called the mzultiplicand. 


FIRST STEPS AMONG FIGURES. 67 


Multiplier. 

The number by which we multiply, (or 
which shows how many times the multipli- 
cand is taken,) is called the szzltzplier. 


Product. 


The result of the multiplication is called 
the product. 


Proof. 


I. Multiply the multiplier by the multi- 
plicand, and if the product equals the pro- 
duct first obtained, the work is probably cor- 
rect. 

II. Or divide the product by the multi- 
plicand, and if the work is correct the re- 
sult will be the multiplier, or divide by the 
multiplier and get the multiplicand, 


~ Multiplication. 
Mueeseeg 2. 42,356)" 3. 35,642 
7 7, 6 


4. 57,463 5.- 35,042 6. 35,246 
7 7 8 


7. 57,463 8. 35,246 9 46,058 
8 6 8 


See Teachers’ Edition, p. 119. 


68 


FIRST STEPS AMONG FIGURES. ‘ 


10. 685,037. II. 485,067 
8 


13. 


12. 857,046 

6 
637,058 14. 364,078 
4 7 


15. 3,640,758 x 8=? 


16. -7,593,04G, gaan 


When there is more than one figure in the 
number by which we multiply, the right 
hand figure of the result is placed under 
the figure we multiply by. 


17. 46,352 18. 56,342 


nd 


20. 
21. 
22. 
23; 
24. 
ons 
26. 
2: 


32 23 


364,526 43=? 
A2: 502 xiha —- 
534,652 x 40=? 
645,362 x 64=? 
63,527 x 76=? 
4675 xO7=7 
57,463 x 74=? 


19. 2,635 
3 


FIRST STEPS AMONG FIGURES. 6g 


28. 357,426x63=? 
29. 47,563 X67=? 
30. 63,527 x 54=? 
31. 352,746 x 34=? 
32. 475,063 x 57=? 
33. 630,574 x 76=? 
34. 57,463 x 35=? 
35. 357,420 47=? 


Division. 


Finding how many times one number is 
‘contained in another is called adzvision. 


Dividend. 


The number which contains the other is 
called the azvidend. (If preferred, the 
number divided is called the adzvzdend.) 


Divisor. 


The number which is contained in the 
other is called the dzvzsor. (If preferred, 
the number by which we divide is called 
the divisor.) 


. OQuotzent. 


The number found by dividing is called 
the quotient. 


7°? FIRST STEPS AMONG FIGURES. 


Proof. 

Multiply the quotient by the divisor, 
and add the remainder if there be one; if 
the work is correct the result should equab 
the dividend. 


Divisor, 7\4321 Dividend. 


Quotient, 617—2 Remainder. 


Short Division. 


(1.) (2.) (3.) 
2)6,846 3)612,921 4)3,281,224 — 


oe 


(4.) (5.) - (6.) 
3)9,182,715 4)2,082,836  5)5,304,535- 


(7.) (8.) (9) 
3)24,152,721 2)128,166,414 4 836,284,820 


(10.) ae (12.) 
3)21,186,912 412,032,828 2)106,416,81& 


See Teachers’ Edition, p. 120. 


FIRST STEPS AMONG FIGURES. 7® 


a 


With remainders. * 


13. 7,372+3=? 

14. 14,173+4=? 
15. 16,427---3=? 
16. 823,975+3=? 
Meeate2,020>4—? 
18. 1,327,178+5=? 
19. 954,540+4=? 
20 5,663,032+3=? 
21.-20,173,385-4=? 
22. 273,823+5=? 
23. 925,588+6=? 
24. 257,429+-4=? 
Bea 741,254--6—? 
26 376,233+5=? 
27. 3,804,168+7=? 
28. 4,873,290+5=? 
29. 4,497,190+7=? 
30. 3,880,916+6=? 
31. 3,146,153+4=? 
32. 172,448+7=? 
33. 20,740,905 +6=? 


Count by 7’s from 7 to 42. 


* Teach the division series with remainders in Teach-- 
ers’ Edition, p. 107, before solving these examples. 


72 FIRST STEPS AMONG FIGURES. 


ADDITION TABLE. 9’s and review. 
C 
ies 518 4g 
9 4 7h One 
107 LL 15 12;1G0 separ 


Saat 
Pe 
15412 


6 9 
4 5 
10141 


b 
6 9 3220 
eee 5 6 


rE 
3 5. St 2°75 ome 
6 9 44967 89 
21I9 13 17 14 12/1113 1014 18 
ie h 
So 37> 4iyen sae 
Ae 35> OF Saas 
OI 3uIO 44s bE 


5 
6 
I 
a e 
8 eG 
9 me 


9 Ae, 
oe 8 
16 I Sit 


7 


“SUBTRACTION TABLE. Q’s and review. 


I I 


8. cere 


C7’ 
®% O 


He 
17, 13:16 526 
9: 87 


8 5 9 63 


1418 9 12 14)17, 812.9013 
89.5 8400918 AS ies 
6 94 8 5194 73 6 


h 
14 1 
5 
9 


See Teachers’ Edition, p. 123. 


FIRST STEPS AMONG FIGURES. 73: 


MULTIPLICATION TABLE. 9’s avd review. 
C 


Ow 
WEI STOVE eo RA O)\ ses 
cn 


DIVISION TABLE. | G's and review. 


a 
30 45 24 


56 OF) 12.3 
79549 4 
8 3 


3 


2 
8 
4 


Oo | OF 


vn Aiwa Oo & 


42 18 35 16 72/40 63 36 72 45 |3 
ps A 9) 8 7 


48 81 49 24 40 18 5/ 
Behe 54 (2-.S 
Oey Ar8 9 6 


See Teachers’ Edition, p. 124. 


8 
6 


Ta. FIRST STEPS AMONG FIGURES. 


1. In a school there are 8 recitation 
‘rooms, each room has 7 benches; how 
many benches in these rooms ? 

2. There were 60 freight cars on a road 
.and 6 of them were destroyed in a collision ; 
_ how many remained? 

3. If a man can draw 7 loads of sand 
in 1 day, how many days will it take him te 
draw 42 loads? 

4. Ina class-room there are 6 seats, and 
-each seat will hold 8 pupils; how many 
pupils can be seated in the room? 

5. Six sheep were put into a flock con- 
taining 87; how many then in the flock ? 

6. Pineapples are 6 cents each; how 
many can be bought for 48 cents? 

Count by 7’s from 7 to 70. 

7. If Nellie pays 5 cents for candy and 
has 8 cents left, how many cents had she 
-at first? 

8. How many legs have 7 cats? 

9. How many quarts in 8 gallons? 

10. How many quarts in 12 pints? 

11. If John buys some candy for 18 
“cents, some peanuts for 7 cents and an 
orange for 6 cents, how many cents does 
-he spend? 


See Teachers’ Edition, p. 123. 


FIRST STEPS AMONG FIGURES. 75 


12. * If 8 sheep cost $32, what will one 
sheep cost? 

13. How many fingers and thumbs have 
© boys? 

14. Willie bought 2 pounds of crackers 
at 8 cents a pound, and half a pound of 
cheese at 14 cents a pound; how much 
money did he spend ?. 

15. Harry had a ten-cent piece, a five- 
cent piece and two three-cent pieces ; how 
much money had he? 

16. Annie had 60 cents; she spent 30 
cents for a doll, and received 10 cents and 
2 more from her mother for doing an 
errand ; how many had she then? 

17. Thomas had 13 chickens and II 
little turkeys; the cat caught 5 of the 
‘chickens and the rats caught 4 of the little 
turkeys ; how many of both were left ? 

Count by 7’s from I and 2 to 71 and72. 

18. William had 7 cents and John had 
twice as many; how many had both the 
boys? 

19. I have 18 pupils in a spelling class; 
4 of them misspell some of their words ; 
how many recite perfectly ? 


*The character $ means dollars and $32 is read thirty- 
two dollars, 


76 FIRST STEPS AMONG FIGURES. 


20. George had 7 glass agates, 8 china. 
and 6 common marbles ; how many marbles. 
did he have? 

21. If 24 apples be equally divided among 
8 boys, how many will each have? 

22. Walter took 56 cents to the store to: 
buy sugar at 8 cents a pound; how many 
pounds could he get? 

23. Katie’s mother gave her g cents, her 
father 7 cents, and her aunt 4 cents; she 
bought 3 oranges at 4 cents each and spent. 
the rest of the money for candy at 2 cents. 
a stick; how many sticks of candy did. 
she get? 


I, 5435 2. 3454 3. 5342 


4354 5323 3455 
3343 2345 5534 
5425 4534 2345 
2342 5453 4453 
3454 3225 5225 
ao a0 4544 3542 
4344 2345 4354 
5432° 5432 5335 
2345 4543 4543. 
4554 1234 3254 
3435 5432 5432 
5342 3555 1234. 
4554 4343 4555 
3343 5443 5321 


See Teachers’ Edition, p. 126. 


FIRST STEPS AMONG FIGURES. 


4. 3452 
5324 
4543 
5435 
3354 


5445, 


4554 
3425 
5432 
2345 
4553 
5434 
4423 
3545 
5434 


6. 4356 
3645 
5234 
6563 
3656 
4345 
5456 
6563 
3625 
5356 

6545 


7. 3564 
6453 
2566 
4035 
5 364 
6532 
3056 
4304 
5635 
6556 
3440 


5. 4532 
3254 
4325 
5543 
3454 
3434 
2345 
5432 
4545 
3454 
5235 
4523 
5445 
4354 
5432 


cea eee 


8. 6345 
2436 
6563 
3656 
5365 
6534 
4655 
5362 
6656 
3545 
6666 


es 


a, 


78 


II. 
12. 
13. 
14. 
I5. 
16. 
17. 
18. 


FIRST STEPS AMONG FIGURES. 


9. 3456 10. 6354 
6563 5665 
5635 6536 
6366 3456 
3556 6365 
6645 6653 
4536 5546 
5663 3665 
6636 6236 
6355 6463 
5663 3554 


684,632—47,757=? 


Ans. 636,875. 


43,120—2,765—? Ans. 40207 


935,043 —77,387=? 
754,231—20,154=? 
836,425 —68,268=? 
364,135 —71,543=? 
463,845 —98,536=? 
763,843 —46,452=? 


Ans. 858,256. 
Ans/725,0774 
Ans. 768,157. 
Ans. 292,592. 
Ans. 365,309. 
Ans. 717,391. 


19. 


79 


——— 


FIRST STEPS AMONG FIGURES. 
6453 20. 3456 = 21. 3645 
3566 6566 6556 
4635 5635 2663 
6326 4364 6350 
5663 5662 5505 
4566 6556 4636 
3656 6636 6353 
6305 3665 5064 
5636 5326 6656 
6656 6553 3533 
5625 3466 6665 
5564 6530 3456 
4563 5402 6363 
22. 5663 23. 6543 
6656 2345 
3546 6234 
6635 5623 
5366 4562 
4653 3456 
6636 6566 
3565 5335 
6366 46606 
5436 5656 
6563 6535 
3666 6666 
6525 3553 


Sce Teachers’ Edition, p. 126. 


80 FIRST STEPS AMONG FIGURES. 


When there is a cipher or ciphers in the 
multiplier between significant* figures, do 
not use it in multiplying, since nothing 
times any number is nothing ; but be care- 
ful to write the first figure of each product 
under the figure you multiply by. 


. 57,364 x 304=? 
- 47,563 x 504=? 
246,75 20700—1 
» O20,A7 5800 7—1 
50,740 x10 7a. 
. 46,375 x 564==? 
P2315 PO3 7320 =e 
. 6,375 X657=? 
NOB 74205 A 
7AT,580 x O8—? 
119647,5 530-05" 
12. 364.758 x 48=? 
13. 105,743 +4=? 
14, 3,176,207 = 5—=¢ 
15. 1,418,407 -4—? 
16. 22,477,527-+-6=? 
17, 2,620,3690+4=? 
18. 23,176,234+5=? 


to 
OO ON AN BW DN 


* The significant figures are 1, 2, 3, 4, 5, 6, 7, 8 and g, 
See Teachers’ Edition, p. 128. 


FIRST STEPS AMONG FIGURES. 81 


19. 30,249,742+4=? 
20 2,625,160+6=? 
Glee 00,850)-7 =? 
22) 934,459,582 -6 =? 

When the divisor is not contained in the 
partial dividend, write a cipher in the quo- 
tient and the partial dividend will be the 
remainder to be prefixed to the next figure 
of the dividend. The teacher will illustrate 
by the following examples: | 

BAe O,4 215 A? 
25. 44,593,072+7=% 
20, e2, 115.085 =5=? 
27. 37.548,510+6:=? 
28. 25,233,2905+7=? 
29. 37,220,621+8=? 
30. 42,519,206+6=? 
31. 40,379,014+7=2 
22, 52,594;387+8=? 
33. 20,326,941+8=? 
I. 874,895 + 20I—? 
250728,550-201=? 
. 12,756,024+ 3,012? 
8,604,338+2,031=? 


*L Ww 


CONT OVM 


FIRST STEPS AMONG FIGURES. 


. 16,154,855 +3,024=? 

. 40,486,701 +2,032=? 

. 2,595,900+4,023=? 

. 1,460,703+403=? 

. 10,357,440+ 3,024=? 

. 14,244,539+3,014-=? 

. 13,822,604+-4,034=? 

. 25,079,486 — 7,036=? 

. 38,968,336+6,035=? 

. 278,881,300 —5,040—2 
. 334,654,184 ~ 70,364=? 
. 349,143,867 +6,057=? 
. 226,396,593 ~60,453=? 
. 18,651,776 —4,076=? 

. 526,026,567+7,046=? 
, 236,326,931 — 5,074=—2 
» 39,361,095 +6,024=? 

. 540,611,445 ~7,068=? 
. 450,299,132+60,378=? 
. 327;040,029 ~ 5,048=? 
- 413,535,490+ 70,485 =? 
. 241,993,12 + 5,086=? 

. 546,695,551+80,574=? 
. 465,406,942 + 7,068=? 
. 289,561,188 +6,075 =? 


See Teachers’ Edition, p. 136. 


30. 


FIRST STEPS AMONG FIGURES. 


4563 
3656 
6366 
5665 
4656 
6546 
6665 
5656 
3456 


6543. 


5655 
6366 
5621 


33. 


31. 5465 32. 4536 

6536 5665 

6653 4354 

3566 6635 

6632 5666 

5 366 6563 

6556 6356 

3465 3065 

6634 5636 

5663 6565 

2556 3663 

6665 6356 

5326 6565 
5634 34. 3456 
6565 6536 
3456 5665 
6563 3566 
4050 6343 
5665 4656 
6363 3565 
5636 6666 
3565 5350 
6366 4663 
5636 6536 
6653 5465 
3546 6323 


83 


84 FIRST STEPS AMONG FIGURES. 


1. A boy had 15 marbles and lost all 
but six of them; how many did he lose? 

2. Mary comes to school 5 days ina 
week; how many days does she come in 
8 weeks ? 

3. There are 55 sticks of candy ina 
jar; if 8 little girls each buy a stick, how 
many sticks will be left in the jar? 

4. How many marbles can a boy buy 
for 27 cents at three cents apiece ? 

5. Fanny had 8 cents and Julia had 9g 
cents, how many did both girls have ? 

When an example involves several opera- 
tions the pupil should give but one at a 
time. 

6. How much more will 6 oranges cost 
at 4 cents each, than 7 peaches at 2 cents 
each ? 

Solution : If one orange cost 4 cents, 6 
oranges will cost 6 times 4 cents or 24 
cents. If one peach cost 2 cents, 7 peaches 
will cost 7 times 2 cents or 14 cents. If 
the oranges cost 24 cents and the peaches 
I4 cents, the oranges cost as much more 
than the peaches as the difference between 
24 cents and 14 cents or I0 cents. 

7. Henry had 25 cents;he gave 3 cents 
each to his brother and sister, spent 5 cents 
for an orange and 2 for candy; how many 


cents had he left ? 
See Teachers’ Edition, p. 137. 


FIRST STEPS AMONG FIGURES. 85 


8. Harvey had a twenty-fivecent piece, 
a ten-cent piece, a five-cent piece and a 
three-cent piece ; how muchmoneyhad he? 

g. If I had 4 apples and found as many 
amore, and ate two of them, what part of a 
dozen had I then? What are they worth 
at 12 cents a dozen? 

10. ] have a clock that strikes every 
quarter hour ; how many times will it strike 
in g hours? 

11 William spent 12 cents, James spent 
one third as many and three cents more; 
show many did James spend ? 

12. There are 2 little dogs passing ; how 
many eyes, ears and feet have they ? 

13. Three men each take three bags of 
wheat to mill, and each bag contained 2 
bushels; how many bushels did the men 
‘take to the mill? 

14. When milk is 6 cents a quart, how 
Many quarts can you get for 42 cents? 

i5. When milk is 4 cents a quart, how 
Many pints can you get for 20 cents? 

16. In a school-room there are 7 rows of 
‘seats, and 6 seats in each row; how many 
‘seats are there in the room ? 

17. A lady made 7 squares of patch- 
work, and her little girl sewed so many that 
“one-half of what both sewed was 10; how 
amany did the little girl sew? 


86 FIRST STEPS AMONG FIGURES. 


18. How many boxes of wafers at 6 cenits: 
a box may be bought for 9 sheets of paper 
at 2 cents a sheet? 

19. How many barrels of apples at $3 a 
barrel can be given for 6 yards of flannel at 
$2 a yard? 

20. How many four-horse teams can be: 
mers from 20 horses? 

Three fields have each 3 trees, Werte: 
see tree are 3 cows; how many cows in 
the three fields ? 

22. A man bought a duck at 9 cents a- 
pound and paid 54 cents for it; how much 
did the duck weigh? 

23. If 6 oranges cost 24 cents, what cost. 
8 oranges? 

Solution: If 6 oranges cost 24 cents, ome 
orange will cost one-sixth of 24 cents, or 
4 cents, and 8 oranges will cost 8 times 4. 
cents, or 32/ cents. 

24. If a boy walks 15 miles in 3 days, 
at the same rate, how far will he walk in 4. 
days? 

25. If it takes 16 yards of cloth for 2 
suits of clothes, how many yards will it 
take for 6 suits ? 

26. If a boy goes 8 feet in stepping 4. 
times, how far will he go in stepping 7 
times ? 

See Teachers’ Edition, p. 138. 


FIRST STEPS AMONG FIGURES. 87 


27. If 3 men can cut Q acres of grain in: 
one day, how many acres can 6 men cut in 
a day? 

Poet it takes 12° buttons for 3 vests, 
how many buttons will it take for 8 vests? 

29. How many yards of cloth at $3 a 
yard can be bought for 4 barrels of flour at 
£€6 a barrel ? 

30. If 3 men can build a wall in 6 days, 
how long will it take one man? 

31. If 5 men can mow a field of grass 
in 10 days, how long will it take one man ? 

32. If 4 men cut 8 cords of wood ina 
day, how many cords will 1 man cut ina 
day ? 

33. If 3 mencut a pile of wood in g days, 
how long will it take one man? 

34 If 3 mowing machines will cut 27 
acres of grass in one day, how many acres. 
will 7 mowing machines cut in one day? 

35. If a boy earn 63 cents in 7 days, 
how much will he earn in 6 days? 

36. How many books at 4 shillings each 
can you buy for 8 dozen eggs at 2 shillings. 
a dozen? 

37. A teamster drew 8 loads of stone 
each day for 7 days; how many loads did 
he draw ? 

38. A boy gained 7 cents by selling a. 
knife for 42 cents ; what did it cost him ? 


58 FIRST STEPS AMONG FIGURES. 


39. William worked 8 hours at 2 shillings 
an hour, and Henry worked 3 hours at 3 
shillings an hour ; how much did both earn ? 

40. Four girls have each 2 hens, and 
each hen has 6 chickens; how many 
chickens have the four girls? | 


Read the following numbers: 


. 750406300. 

. 45706000043 

. 860000307. 

. 15000045001. 
- 7845078437. 
37147415006. 
47583000000. 
370015 300. 

. 40000036700. 
7 LOOOIOO00O. 


OO ON ANALWN 


Write in Arabic the following numbers : 
1,, Fifteen mil tenth three 
2. Two hun. eight bil. one hun. th. 
3. Three bil. twenty mil. six. 
4. Thirteen bil. nine th. seven hun. 
‘forty-five. 
5. Ninety-one bil. one mil. one th. one. 
6. Four bil. seven hun. fifteen. 
7. Two hun. sixty mil. 
8. One bil. three hun. sixty mil. two 
‘hun, th. 


See Teachers’ Edition, p. 140. 


FIRST STEPS AMONG FIGURES. 8q 


9. Five mil. ninety. 
10. Write in Roman nine hun. thirty- 


11. Write in Roman seven hun. forty-six. 
hematite in. Arabic DCCCXCVII. 

13. Write in words 709460371000. 

14. Write in Arabic ten th. thirty. 


Teach the pupils that the figure at the 
right expresses units of the first order, the 
next figure to the left, units of the second 
order, the next figure, units of the third. 
order, and so on. 


15. Write 7 units of the 5th order, 4 of 
the 3d and 1 of the Ist (in one number.) 

16. Write 3 units of the 8th order, 5 of 
the 7th, 9 of the 3d and 4 of the 2d. 


It may aid the pupil in solving the follow- 
ing examples, to put small numbers in the 
place of the ones given, and see how it 
would be solved without the slate, then 
solve in the same manner. 

Simple problems for the slate, involving. 
Addition, Subtraction and Multiplication : 

1. Mr. Rogers had 746 bushels of wheat 
and sold 197 bushels of it; how much had‘ 
he left ? 

2. George had 295 cents and his father 
gave him 75 more ; how many had he then ? 


“gO FIRST STEPS AMONG FIGURES. 


3. Mr. Smith had 96 bushels of oats 
-and Mr. Jones had 9 times as much; how 
many bushels had Mr. Jones? 

4. Lewis has 87 marbles and John has 
just as many; how many have both boys? 

5. Mr. Howard drew 8 loads of oats to 
market, and there were 79 bushels in each 
load ; how many bushels did he draw to 
market ? 

6. From the sum of 79 and 268, take 158. 

7. How much will a teacher’s salary 
amount to in 14 years, at $875 a year? 

8. James has 47 marbles less than John, 
and John has 174; how many has James? 

g. John lost 15 cents by selling his 
knife for 90 cents ; what did it cost? 

10. Miles took 2341 steps in going to 
school. and Marcus took 560; how many 
more steps did Miles take than Marcus ? 

11. Mr. Decker borrowed $150 and paid 
$65 of it; how much does he still owe ? 

12. What will 46 bushels of barley cost 
at 167 cents a bushel ? 

13. A clerk received a salary last year of 
$1000. He spent $260 for board and $378 
for clothing and other expenses ; how much 
money did he save? 

14. There are 30 days in June, and 31 
each in July and August; how many days 


4n these three summer months ? 
See Teachers’ Edition, p. 143. 


FIRST STEPS AMONG FIGURES. gI 


15. There are 168 acres in Mr. Fox’s 
farm, and Mr. Norton's farm contains 8g 
acres more than Mr. Fox’s; how many 
acres in both farms ? 

16. A man put $950 in the bank; he 
drew out $78 at one time, $45 at another, 
and $159 at another; how much had he 
left in the bank ? 

17. John had 39 marbles and Ezra had 13 
more than twice as many; how many had 
Ezra? 

18. Mr. Brown bought a farm for $8460 
and sold it for $10380; how much did he 
gain? 

19. What cost 369 bushels of oats at 68 
cents a bushel? 

20. I have 216 bushels of potatoes in 3 
bins ; there are 59 bushels in one bin anc 
98 bushels in another; how many bushels 
in the third bin? 

21. A boy having 85 cents, bought a top for 
18 cents, aball for 25 cents,andsome oranges 
for 27 cents; how many cents had he left? 

22. If there are 76 bushels of corn ina 
bin that will hold 950 bushels, how many 
more bushels of corn may be put into it? 

23. A farmer filled at one time 29 bags 
with oats,and at another 47 bags. If he 
put two bushels in each bag, how many 
bushels of oats were put in all the bags ? 


92 FIRST STEPS AMONG FIGURES. 


24. From one million eight hundred 
thousand take fifteen thousand ninety. 


1. 4507 2. 7045 3. 4756 


7654 5774 3567 
6347 6547 6635 
7576 5732 7456 
4757 7665 6573 
5674 4576 5746 
7505 5766 7455 
4757 7537 6567 
5664 6452 7730 
7575 3776 6564. 
2346 6565 6475. 
4. 7546 5. 7654 
4757 6537 
5675 5465 
6757 77506: 
7577 4675. 
4664 3456 
7357 6747 
5675 7504. 
4567 6475. 
7654 7746 
4735 4653, 


See Teachers’ Edition, p. 144. 


FIRST STEPS AMONG FIGURES. 
Co 


6. 3456 
7653 
6575 
5747 
7356 
5676 
4747 
5635 
4567 
7476 
5745 
6574 


7. 6754 
5673 
7566 
4375 
7047 
5774 

6757 
7577 
4652 
5767 
7760 
5473 


8. 5746 
7457 
6575 
7067 
5734 
4757 
7676 
4567 
6735 
5657 
4576 
7465 


93 


FIRST STEPS AMONG FIGURES. 


II. 4,362,516—754,359=? 
129.730.0522 7S .07 28 

13. 642,534—26,356= ? 

14. 352,432—86,354=? 

15. 6,425,314—374,321=? 
16. 463,524—71,876= ?. 

17. 425,362—17,654= ? 

18. 364,253—86,174=? 

19. 4603,521—186,357? 

20.. 483,654—91,087=? 
21.:635,245—12,567? 

22. 837,524—259,280= ? | 
23. 43,452,431 —4,238,865 =? 


24. 756,324—85,543=? 


25. 4,738,536—973,659= ? 
26. 86,357—7,269= ? 

27. 34,023—0,876=? 

28. 45,300,435 —2,430,520= ? 
29. 74,200,032—5,140,054= ? 
30, 43,250,001 —5,726,025=? 
31. 3,400,564—210,739= ? 


32. 45,700,325 —8,730,153=? 


I. 796,845 x 89=? 
22 cA Toma Sete 


FIRST STEPS AMONG FIGURES. 


. 68,975 x 79=? 

. 647,583 x 467=? 
. 68,574 456=? 
bASO,075 X S07? 
59,007. x O64—? 
BOO 7 07.5 = 2 

- 4,790 x 39=? 

. 85,974x 84= 2 
070,035 x Go= ? 

. 68,974 x 79= ? 
AO, / 00% OF fo 

. 59,068 x 604= ? 
170,008:x 907=? 
ee7O; 200 x 7OO='¢ 

. 860,479 x 709= ? 
. 759,068 x 407? 

. 47,096 x 609= ? 

. 748,609 x 507= ? 
, 58,709 x 608= ? 

. 00,047.x 7090= ? 
PO7,500:< 507— ? 

- 79,689 x 4,759= ? 
. 896,748 x 6,978=? 
7,500,005 — 1,25.1,:2908= ? 


See Teachers’ Edition, p. 145. 


95 


to 
\O 


— = me 
Ne OW ON AM AW WN 


QO = = = = = = 
00 ON AwM DB W 


NS 
tt 


FIRST STEPS AMONG FIGURES. 


- 3,740,683 —923,754= 
725 O20 520 ee 
: 7,430,246 —7,503,472= ? 


. 3,241,402+7=? 

. 5,079,005 = 62 
3,802,457 O=-0 

. 49,167,544+9=? 

. 5,076,335+9=f 

» 3,372,081 +9=? 

. 3:725,801+8—2 

. 52,301,166+8=? 

. 48,0904,605+-7=? 

. 43,719,125+9=? 

. 52,620,186+8=? 
(40,224,713 eet 
507,21 7710 Le sat 

. 608,226,845 -9=? 
. 37,100,695 +8=? 

. 460,241,323+7=? 

. 4,781,158,859+6=? 
. 4,885,157,701+7=P 
. 389,15 8,560+6=? 
. 5,240,869 -7=? 


» 3,356,977+7=? 
See Teachers’ Edition, p. 146. 


FIRST STEPS AMONG FIGURES, 97 


2G, 727,741 Oo? 

Mpst 5.271 Ss! 

. 21,307,398+5,024=? 

. 3,722,901+607=? 

. 14,204,241 ~403=? 

. 158,032,783 = 30,135? 

. 38,693,395 —6,024=? 

. 38,464,305 +50,396=? 

. 31,970,764~7,048=? 

. 3,891,687,541+60,475= ? 
. 2,925,490,533+60,479= ? 
. 462,857,740+8,069= ? 

. 2,934,401,497+70,586= ? 
. 226,663,766+60,379= ? 

- 194,513,933+40,297—? 
$2184,110,003=-5,048= ? 

700. 701s5d0+6,037= 7 

. 32,688,027,778+70,486= ? 
. 674,.476,820+9,037=? 

. 27,971,095 +6,074= ? 

. 355,212,265+5,030=? 

- 23,377,790+5,024= ? 


452,113,508 +7,056= ? 


. 299,019,935+406= ? 


98 


FIRST STEPS AMONG FIGURES. 


. 2,408,592,665 +5,064=? 


47. 36,979,544 8,026= ? 
48, 426,696,721 —7,01d=" 
49. 437,791,338+6,038= ? 
50, 2106,526,004-+-5,027=? 
51. 2,464,085,695 + 604,978 ? 
52. 238,049,090+ 5,037= ? 
53. 4,110,929,380+70,586= ? 
54. 283,899,778+6,034= ? 
55. 4,980,403,784+ 70,496= ? 
56. 52;202)15 Sea 

57. 251,864+361=? 

58. 434,801+573=? 

59. 268,500+463=? 

60. 463,799+582—=! 

61.* 356, 116— 365=? 

62. 357,243+465=? 

632: AO3j12 oe Sober 

C4. 414,017+473=? 

65. 4,174,696+485 =? 


* When the left hand figure of the divisor is equal to- 
the left hand figure of the dividend, if the next figure of 
the divisor be greater than the next figure of the divi- 
dend, point off as if the left hand figure of the divisor 
were greater. The divisor (in one step of the opera- 
tion) never is contained more than nine times. 


See Teachers’ Edition, p. 148. 


FIRST STEPS AMONG FIGURES. 99 


66. 5,499,513+796=? 
67. 5,538,824+684=? 
68. 7,084,249+896=? 
69. 30,734,480+645=? 
70. 43,722,966 573=? 
71. 270,578,240 4,035 = 
Bae 2005430, 50 > 4,657== 


1. William paid 54 cents for 6 doves; 
what did each dove cost? } 

2. In an orchard there are 6 rows of trees 
and 7 trees in each row; how many trees 
in the orchard ? é 

3. Henry has 8 cents in one pocket and 
9 Ronis in the other ; how many cents has 
he? 

4. James has 8 apples and his sister has 
6; how many more has James than his sis- 
ter? 

5. What cost 9 knives at 7 shillings each ? 

6. A boy paid 25 cents for a ball and sold 
it for 18 cents ; how many cents did he lose? 

7, If 1 pencil cost 4 cents, what will 8 
pencils cost ? 

8. George bought a knife for 8 shillings, 
a ball for 5 shillings, and a bat for 2 shil- 
lings ; what did he pay for all ? 

9g. Marcus spent 8 cents for lemons at 4 


cents each; how many lemons did he buy? 
See Teachers’ Edition, p. 150. 


e 


[00 FIRST STEPS AMONG FIGURES. 


10. If 5 men cut ten cords of wood ina 
day, how many cords will 7 men cut? 

11. If 2 men can dig a certain ditch in4 
days, how long will it take one man to dig 
it ? 

12. If 4 men can cradle 12 acres of grain 
in one day, how many acres will one man 
cradle in a day? 

13. How many rods of wall will one man 
build in a day, if 3 men build*g rods in one 
day? 

14. If 3 boys can pick the stones from a 
meadow in 9 days, how many days will it 
take one boy to pick them ? 

15. How many weeks in 35 days? 

16. If 4 pounds of sugar cost 36 cents, 
what cost 8 pounds ? 

17. When a pineapple costs 18 cents and 
an orange costs 6 cents, how much more 
does the pineapple cost than the orange ? 

18. If a boy can walk 12 miles in 4 
hours, how far can he walk in 5 hours? 

19. If 4 men can do a piece of work in 
8 days, how long will it take one man? 

20. Jane bought 5 figs for 3 cents each, 
anda yard of cloth for g cents; how much 
did she pay for all? 


See Teachers’ Edition, p. 151 


FIRST STEPS AMONG FIGURES. IOl 


21. Mary sews 4 hours each day, how 
~-many hours does she sew in a week ? 

22. How much will a man’s board for a 
aveek cost at 4 shillings a day ? 

23. How much will a man earn in a 
week, if he gets 9 shillings for a day’s work ? 

24. If 15 cats are on a wall and every 
third cat jumps off, how many are left? 

25. There are 8 quarts in a peck, how 
many pecks in 32 quarts? 

26. How many quarts in 3 pecks? 

27. A boy picked 16 quarts of beans and 
‘sold them at 25 cents a peck; how much 
money should he receive ? 

28. Charles has 7 cents and his brother 
3 more than twice as many ; how many 
have both? 

29 On Monday morning Mary had 20 
‘sticks of candy; she ate 2 each day, how 
‘many had she left the next Monday night ? 

30. Arthur had 11 peaches, he ate 3 
and gave his sister half of the rest; how 
anany did he keep? 

31. How many marbles, 2 for 4 cents, 
«an you get for 18 cents? 


102 ' FIRST STEPS AMONG FIGURES. 


EXAMPLES FOR THE SLATE. 


If the pupil will use small numbers in- 
stead of the large ones in the following 
examples, and think carefully how he would 
solve them if they were not for the slate, 
and then do the same with the numbers. 
given, using the slate as a help, he will be 
Bede ascisted: 

A man had $3,210, he spent $978 for 
ee and $749 for corn ; how much money: 
had he left ? 

2. What is the product of 9,687 and 75 ? | 

3° From the sum of 3796 and 4279, take 
their difference? 

4. If a farmer have 256 gallons of cider, 
how many barrels holding 36 gallons can he 
fill ? 

5. Froma cistern holding 743 gallons, 98° 
gallons were pumped out and afterwards 39. 
gallons poured in; how many gallons were 
then in the cistern ? 

6. What cost 37 carriages at $185 each ? 

7. If a ship sail 7289 miles in 37 days,. 
how many miles does she sail per day ? 

8. A miller paid $169 for 78 bushels of 
wheat, $97 for oats and $395 for corn 3. ° 
what did he pay for all of the grain? 

g From the sum of 397 and 6708, take 
69. 

% See Teachers’ Editon, p. 152. 


FIRST STEPS AMONG FIGURES. 103 


10. The difference between two numbers: 
is 347, and the less number is 79, what is. 
the greater number ? 

11. A man died leaving $5600, of which 
he gave his wife $2,800, his son $900, one 
daughter $850 and the rest to another 
daughter ; how much did the second daugh-. 
ter receive? 

12. A man bought 75 sheep at one time, 
and 169 at another ; he sold 86 of them to 
one man and 49 to another; how many 
had he left ? 

13. Mr. Wilson bought one house for 
$4150, and afterward another for $3750 ; he 
sold both of them for $7000; did he gair 
or lose, and how much? 

14. There is an orchard consisting of 24 
rows of trees, and 36 trees in each row; 
how many apples in the orchard, allowing 
an average of 2079 on a tree? 

15. A man owing $7165, gives in pay- 
ment 39 cows valued at $48 each and $750 
in money ; how much does he still -*¢? 

16. Add 16 thousand 20, fifty millio. *t, 
thousand nine, 79 thousand 847, and 9 m - 
lion 79 thousand 8." 

17. How many tons of hay at $18 a ton 
must be given for 16 horses at $153 each? 

18. 6394+91,758+9,347 481,731 $9,342 
+ 35,4464 8,237+12,849+87,677=? 


104 FIRST STEPS AMONG FIGURES. 


19. A grocer spent $881 for molasses 
and sugar; he paid $368 of the money for 
molasses, and the rest for 27 barrels of 
sugar; how much did the sugar cost a 
barrel ? 

20. How mary yards of cloth in 68 bales, 
each bale having 97 pieces, and each piece 
containing 29 yards? ; 

21 Paid $6 each for 75 sheep, and sold 
the flcck for $400; did I gain or lose,and 
how much? 

22. How many horses at $165 each can 
he bought for $2360? 

23. How much is gained by buying 48 
cows at $37 each, and selling them at $45 
each? 

24. Mr. Dixon has 225 acres of land 
worth $97 an acre, and Mr. Taft has 196 
acres worth $79 an acre ; how many acres 
have the two together, and what is the 
value of the whole? 

25. A man sold a farm of 96 acres at $9 
ar e-:<, and with the money received for it 
» 2nt a farm of 135 acres; what did he 
yay an acre for the latter farm? 

26. A teacher had his life insured for 
$2500. At the time of his death he owned 
a house and lot worth $1850 and furniture 
worth $475. He owed debts to the amount 


FIRST STEPS AMONG FIGURES. 105. 


of $369; how much did he leave his. 
family ? 

27. There are 5280 feet in a mile; how 
many feet in 709 miles? 

28. A man starts from New York on 
Tuesday morning and travels at the rate of 
57 miles a day; another starts from the 
same place Wednesday morning and follows 
on at the rate of 69 miles a day; how far 
apart are they Thursday night? 

29. James sold a grocer 96 eggs at 15 
cents a dozen, and received 120 cents ; how 
much does the grocer still owe him ? 

30. If there were 365 days in each year, 
how many years would there be in 31390 
days? 

31. Add seventy million nine hundred 
thousand, two hundred six thousand eight 
sixty thousand sixty, seven thousand nine 
hundred, ten million ten thousand ten, and. 
seven hundred fifty-nine million two hun 
dred thirty thousand. 


106 


FIRST STEPS AMONG FIGURES. 


- 7057 


4775 
7777 
3456 
6735 
5677 
7750 
3457 
6574 
5767 


4575. 


ae 
5676 


2345 


4. 4756 


7577 
6645 
7734 
5675 
6756 
7497 
4575 
5047 
7777 
6452 
3567 
6776 
5643 


2, 2456 
5707 
7475 
4567 
7756 
5647 
6775 
7564 
4677 
3450 
5797 
7534 
6467 


3. 7567 
4756 
3675 
6777 
7546 
5735 
6767 
7476 
3456 
3456 
7763 
4567 
2475 


5. 7654 
4775 
54607 
7556 
6775 
3457 
5747 
7674 
6577 
7754 
TFT, 
7777 
4564 
5675 


See Teachers’ Edition, p. 153. 


6. 4508 
3785 
8678 
5786 
8678 
8888 
8888 
5678 
8765 
4876 
5487 


FIRST STEPS AMONG FIGURES, 


107 


7, 6758 8. 8765 
8547 4658 
7808 7777 
8778 3456 
8888 6778 
8888 7507 

3576 8878 
6758 7797 
8087 8585 
5678 UY Ufos 
6786 3456 
10. 3748 
8675 
7887 
4567 
8888 
8888 
4507 
PLAT 
ZELE 
4505 
8486 


108 


FIRST STEPS AMONG FIGURES. 


II. 6758 
7584 
4676 
8778 
3456 
8888 
88838 
6547 
3754 
6678 
7563 
5835 


14. 4786 
8657 
6578 
7865 
4576 
8888 
3456 
ewer 
8765 
3578 
8657 


4768 


12:, 7658) 9 1 Stegone 
4576 4578 
6785 8057 
5467 6785 
8878 8888 
4057 8888 
8765 3450 
6578 7777 
7857 5678 
5686 8753 
8578 4576 
6785 8687 

15. 7684 
4578 
3456 
8888 
8888 
7654 
3567 
WK: 
5648: 
7385 
4637 
8386: 


See Teachers’ Edition, p. 154. 


16. 


FIRST STEPS AMONG FIGURES. 


Book | 1A 4037, 18> 4587 
8546 7584 8635 
7685 8758 6754 
4868 5875 8888 
5784 8888 8888 
FATE 8888 6754 
6548 6753 7578 
3675 4584 4785 
8888 it | 1 ge Vf 
8888 5678 5768 
5678 8765 8654 
8563 3857 4585 
5785 6586 7848 
8678 8465 5686 
19. 8476 20. 7586 
5768 4767 
4567 8658 
TEAM 4875 
sede Fe 8888 
8654 8888 
3568 7654 
8888 3867 
8888 TIT 
4567 4685 
8765 8568 
4478 ERG 
7586 4825 
8357 8674 


£10 FIRST STEPS AMONG FIGURES. 


. 4,570,365 — 323,456=2 
- 9,374,056—636,587=! 
685,700,305 — 296,314,537 =? 
. 76,400,235 — 3,234,567=? 
. 38,500,684—8,769,876=? 
. 7,460,683 —370,876=2 
. 375,600,735 —83,735,829=? 
. 83,640,574—5,71 2,053 
- 794,600,435 —63.732,367=? 
~ 74,300,375 —5 3,620,547 = 
». 8,750,043 —070,230=2 
: 6,7 3,021 O07 3.21 eat 
1°48; 300.5 03 0, COO 7 ==5 
. 487,500,564—65,730,637=? 
. 756,000,375 —85,203,456=2 
. 6,847,0C0,346— 367,020,654=? 
79,068 x §58=? 
80,479 x 74=? 
A NSS S808 ae 
968.570 «FOS —=c 
- 79.689 x 4,759=? 
- 79,807 *6,507—? 
VOSGosocel Oe 

49,678 x 9,876=? 
. 497.896 x 8.659=? 


See Teachers’ Edition, p. 154: 


eit ON AM PW HN & 


at 
NO = 


Oe ee ee | 
00 ON Aw L&W 


by NY NN OY 
Oa) aS oy er 


FIRST STEPS AMONG FIGURES. Ill 


Since moving a figure one place to the 
left increases its value ten fold, and moving 
it two places, ten times ten fold or one 
hundred fold,—to multiply any number by 
10, 100, 1000, &c., annex as many ciphers 
to the multiplicand as there are in the 
multiplier. 


379 X 100=37,900 Ans, 
B02 7 805 %.10,000=7 


Cas 7210! 
pee 65 < 100? 
201 0;320 x 1000=? 
30 875 < 100,000=? 


When there are ciphers at the right of 
either the multiplier or multiplicand, or of 
both, place the multiplier under the miulti- 
plicand so that the significant figures 
farthest to the right shall come under each 
other. After multiplying by the significant 
figures and adding, write as many ciphers 
at the right of the product as there are at 
the right of the multiplier and multiplicand 
together. 


(These directions are given very minutely 
but are not to be committed to memory.) 


II2 


FIRST STEPS AMONG FIGURES. 


34200 
34000 


1368 
1026 


I, 162,800,000 


. $337 50 440 000-5 
46,300 X25 O==¢ 

. 635,000 X 700=? 

. 27,500 X 680,000=? 
. 586,000 x 7,400=? 
400% BOy700—t 

. 6,840 x 7,500? 


8,609 x 800=? 


. 67,900 x 870=? 

. 8.690 x 4,700=? 

. 480,600 X 7,090=? 

. 70,580 x 6,408,000=? 

. 706,900 X 5,078,000=? 
. 68,090 x 70,900=? 

. 640,980 x 10,000=? 


I. 43,188,278+9=? 


7OT,O71, 017-0 ak 


FIRST STEPS AMONG FIGURES. 


0, 150)712,035--3=? 
. 67,815,232+7=? 

GO 20;260,026—.9 8 
295 2:717;437-=- 8 =? 
BEOtO08,27 25507 = O—=? 
fei, 202;100= O—7 

- 49,347,765+7=? 

- 44,160,343+9=? 

+ 5575793,570+7=! 
. 7,664,063,843+8=? 
. 8,044,185,607+9=? 
. 491,093,400+7=? 
5522557410? 

. 520,762,735+7=? 
97230,;2453=? 
S027 4;817'- O—? 


710,863,843+8=? 


BeriOps,.17~ 7! 

. 30,884,751 +7,058=? 

. 34,600,073 +6,032=? 
w1,890,186-5,178=? 

. 1,203,161,896+8,169=? 


5,279,490 814=? 


MeO 20.220 725-7 


See Teachers’ Edition, p. 155. 


113 


Ris FIRST STEPS AMONG FIGURES. 


27. 92,197,3604+6,257—0 
28... 7:038, 402-3430 se 
29 511,764,908— 7,461=? 
30. 946,526,656 -6,397=? 
31. 65,790,555 : 6,847=? 
32°C, 303,000 =7400=ar 
33. 18,827,247-+378=? 
34. 526,493,286 3,794=? 

Since moving a figure one place to the 
right diminishes its value ten fold, and 
two places, ten times ten fold or one 
hundred fold,—to divide any number by 
10, 100, 1000, &c., cut off by a vertical 
line as many figures on the right of the 
dividend as there are ciphers at the right 
of the divisor. 

The number at the left of the vertical 
line will be the quotient, and the number at 
the right of it the remainder. 

Illustration : 78634+100=? 

Solution: 786 | 34 the quotient is 786 
and 34 is the remainder. 


I. 793,468 10,000=? 

. 37,680 = 100=? 

. 2,347,600+ 100,000=? 
76,210,048+ 100 =? 

. 372,938,641 + 10,000,000=? 


uy B WwW WN 


FIRST STEPS AMONG FIGURES. Il5 


To divide by any number with ciphers at 
the right. 


78,673 700=? 


Divide both dividend and divisor by 100, 
and cutting off the 2 figures at the right, and 
the example becomes— 


7|00)786|73 
I12—273 rem., or 
273 
Il 2—— 
700 


Divide the number at the left of the 
vertical line in the dividend, by the number 
at the left of the vertical line in the divisor, 


and to the remainder annex the figures of 
the dividend cut off. 


45|000)612|370(13 quo, 
45 


162 
135 


———e 


2IRTOTeM: 


£16 FIRST STEPS AMONG FIGURES. 


Hence the rule: To divide by any num- 
ber with ciphers at the right, cut off the 
ciphers at the right of the divisor by a 
vertical line, and also as many figures at the 
right of the dividend. Divide the remain- 
ing number in the dividend by the remain- 
ing number in the divisor, and to the 
remainder annex the figures cut off from the 
rightof the dividend for the true remainder. 


. 18,228,211+37,500=? 

. 5,142,762,131 - 750,000=? 
. 546,927,300+687,000=? 
976,514,152 >-706,500-—¢, 

. 8,734,758+ 10,000==? 
350,870,000+ 3,580,000=? 
» 3,278,300 7,000—¢e 

. 87200 X22 000—2 

. 7,162,323 ~QoO=? 

. 394,690,750~ 5,800=? 

. 20,85 0,010=> 3;750—2 


=~ OO OCNANAWDN = 


— 


12. From six billion six thousand six, 
take eighty million eight. 
13. 27,752,320,172~5 70,000= P 
I4. 2,910,144,700+ 36,800= P 
See Teachers’ Edition, p. 155. 


15. 
16 


Li 
18. 


FIRST STEPS AMONG FIGURES. I17 


178,576,495 + 100,000=? 
365,820,038 — 6,000=? 
475,308,056 — 48,600=? 


Subtract forty-five million, one 


thousand ten, from forty-two million seven 
hundred thousand. 


19. 


20. 
oe 
oan: 


23 
24 


441,937,000+597,000=? 
30,500,857,231-+3,780,000=? 
B05 7 200 000-1 

47,096 x 8,600=? 
70,286,631+900=? 

7900 x 10O=? 


When the multiplier is less than 13 the 
pupil should be taught and vreguzred to 
multiply but once through, multiplying by 
II or 12 as he has already been taught to 
multiply by 4, 5 or 6. 


An Bw nN 


pvo;007% 12—? 


69,789 x I1=? 
545030 x 12=2 
845.768 x 12=? 


. 68,094,796 x II=? 
m5oO,074 x. 12—? 


See Teachers’ Edition, p. 157. 


118 


FIRST STEPS AMONG FIGURES. 


. O579,087-K1 22 
60/487,005 412-28 
10;408;5 70 01 face 
. 479,058 x 12=? 

; 07,500 04 7aale ae 
1-0,075.007eele ae 
2-69;786:005 XA2—6 
. 94,079,689 x 12=? 
. 979,806 x 12=? 
2-085. 700 4 tee 

, 8:798,070 x22=? 


49,897,697 X 12=? 


. 97,987,986 x I2=? 

| 7. O8Q.O85 20Igat 

. 1,185,401,377-—- 12s 
(540,500.40 Loa iler 
. 820,499,872+12=? 

. 1,044,922,315+12=? 
. 9,438,575,909 + 12=? 
. 9.568,867,509+11=? 
. 455,976,730+12=? 

. 956,219,025 12=f 

. 898,436,885 + 12? 

. 7,434,306,992 ~ 11=? 


FIRST STEPS AMONG FIGURES. II9g 


mr SSSSSSEESeeSeSSesesSsS 


FOR ORAL RECITATION, 


I. A boy having 25 cents, bought mar- 
bles at 4 cents each, keeping 5 cents of the 
money ; how many marbles did he buy ? 

2. Jane lost 10 cents on her way to the 
post-office, and spent the rest of her money 
for 10 3-cent stamps; how much money 
had she when she started ? 

3. * Mark earned 8 cents, lost 5 cents, 
and then found 10 cents, when he had 25 
cents; how much money had he at first ? 

4. What cost 12 pounds of sugar if 7 
pounds cost 63 cents? 

5. A man spent $5, then earned $7, 
and after giving away $6 found he had $15 ; 
how many dollars had he at first ? 

6. If 6 apples cost 2 cents, what cost 18 
apples ? 

Solution: If 6 apples cost 2 cents, 18 
apples, which are 3 times 6 apples, will cost 
3 times 2 cents or 6 cents. 

7. If 4 marbles cost 3 cents, what cost 
24 marbles? 

* If he had 25 cents after finding 10 cents, before he 
found it he had the difference between 25 cents and Io 
cents, or 15 cents. If he had 15 cents afer losing 5 
cents, defore he lost it he had the sum of 15 cents and 5 

cents or 20 cents. If he had 20 cents after earning § 
cents, before he earned it he had the difference between 


20 cents and 8 cents or I2 cents. 
See Teachers’ Edition, p. 161. 


120 FIRST STEPS AMONG FIGURES. ~ 


8. What cost 20 figs, if 5 figs cost 2 
cents? 
g. If 3 oranges cost 12 cents, what cost 
7 oranges ? 
10. What cost 12 lemons, if 5 cost 25 
cents ? 
11. How many apples can be bought for 
15 cents, at the rate of 5 for 3 cents? 
12. If 3 oranges cost 10 cents, how many 
may be bought for 40 cents? 
13. 30 cents will buy how many apples, 
at 9 for 6 cents? 
14. How many figs may be bought for 
24 cents, at the rate of 3 figs for 2 cents? 
15. If 4 marbles cost 5 cents, what cost 
20 marbles? 
16. If 5 lemons cost 20 cents, what cost 
9 lemons? 
' 17. What cost 30 pears if 3 pears cost 5 
cents? 
18. If 4 peaches cost 3 cents, what will 
24 peaches cost ? 
19. At the rate of 2 oranges for 9 cents, 
how many may be bought for 18 cents? 
20. If 3 men cut 6 cords of wood in a 
day, how many cords will 7 men cut ina 
day? 
21. If 3 men dig a ditch in 12 days, how 
long will it take one man? 


See Teachers’ Edition, p. 162. 


* FIRST STEPS AMONG FIGURES. 1 | 


22. If 4 men harvest a field of wheat in 
8 days, how many days will it take 1 man 
to harvest it ? 

23. If 12 men can dig a field of potatoes 
in 13 days, how many men will do it in I 
day? 

24. How many men can load a car in I . 
hour, if 2 men can load it in 4 hours ? 

25. If 4 men can doa piece of work in 
12 days, how long will it take 3 men to do 
it? 

26. How many days will it take 6 men 
to earn $32, if it takes 4 men 6 days to 
earn it? | 

27. If 4 men can doa piece of work in 
9 days, how many men can do it in 6 days ? 

28. If 6 men can doa piece of work in 
4 days, how many men will it take to do 
the work in 3 days? 

29. If 6 men can doa piece of work in 
12 days, how long willit take 4 men todo it ? . 

30. If 4 men can build 12 rods of wall 
in a day, how many rods can 6 men build 
in a day ? 

31. How many men will build a wall in 
12 days, if 6 men build it in 8 days ? 

32. If 3 men cut 7 cords of wood in a 
day, how many cords will 12 men cut ina 
day? 


122 FIRST STEPS AMONG FIGURES. 


°33. A gitl took 7 pins from a paper and 
then put on 9; her brother afterwards took 
off 6, and left in it 24; how many on the 
paper at first? 

34 How many men will do a work in 6 
days that 9 men do in 4 days? 

35. If 8 men do a work in 6 days, how 
many men will do it in 12 days? 

36. If 4 men do a work in 12 days, how 
long willit take 6 men? 

37. If 3 pencils are worth 11 cents, how 
many pencils can be bought for 33 cents? 

38. A girl having a paper of candy, ate 
7 pieces ; then her brother gave her § pieces, 
atter which she gave her mother g pieces. 
She had left 27 pieces ; how many pieces 
had she et first ? 

39 What number divided by 2 will give 6? 

40 If 4 cords of wood cost $20, how 
many cords can be bought for $35 ? 

41. If 6 vests are worth $24, what are 
Q vests worth? 

42. If 5 cords of wood cost $24, what 
will 15 cords cost ? 

43. What number divided by 3 will get 
Loe 

44. At 10 cents a pint, what willa gallon 
of molasses cost? 

45. How many bushels of potatoes at 4 


FIRST STEPS AMONG FIGURES. 123 


shillings a bushel may be bought for 3 
‘bushels of wheat at 12 shillings a bushel ? 

40. A boy gave 8 marbles worth 2 cents 
apiece, for 7 pencils worth 3 cents each; 
how much did he gain? 

47. How many eight-gallon cans will be 
required to hold 56 gallons of milk? 

48. If 15 bushels of wheat will make 3 
barrels of flour, how many bushels will 
make 8 barrels? 

49. How many yards of cloth at $6 a 
yard will pay for 9 tons of coal at $8aton? 

50. When flour is $7 a barrel, how many 
barrels can be bought for $8, and 9 bushels 
of wheat at $3 a bushel? 

51. 96 eggs are how many dozen? 

52. If 8 horses eat 48 bushels of oats in 
2 weeks, how many bushels will 5 horses 
eat in the same time? 

53. Our school has a recess in the fore- 
noon and also in the afternoon. If there 
are one hour of school before each recess 
and two hours after each recess, how many 
hours of school in a week? 

54. If two apples cost one-half of 10a 
cents, how many can be bought for 15 cents? 

55. How many three-cent stamps can be 
‘ought for 27 cents? 

56. A boy caught some fishes ; he threw 


(24 FIRST STEPS AMONG FIGURES. 


away 7, then caught 3 and bought 2, when 
he had 14; how many did he catch at 
first ? 

57. Kate lives two miles from school, 
and does not go home at noon; how far 
must she walk in a week if she loses no 
time at school ? 

58. Frank has 6 five-cent pieces, 4 three- 
cent pieces and five two-cent pieces; how 
many cents has he ? 

59. John has 7 cents, his brother 8, and 
.their sister has 4 more than both of them ; 
how many have they all ? 

60. Which costs the more, 3 lemons at 
4 cents each or 6 pears at 2 cents each? 

61. A boy went to the grocery with 25 
cents, and bought 2 pounds of sugar at 9 
cents a pound ; how much change should he 
bring back if he has 2 cents for doing the 
errand ? 

62. If 2 barrels of flour will last 3 men 
6 months, how long will it last 9 men ? 

63. Bought some peaches for 24 cents, 
at the rate of 5 for 2 cents, and divided 
them equally among 6 boys; how many 
did each boy receive? 

64. If 7 bushels of clover seed are worth 
$42, how many bushels of wheat at $2 a 
bushel will 3 bushels of clover seed buy ? 


FIRST STEPS AMONG FIGURES. 125 


65. If 2 men start from the same place 
and travel in the same direction, one 6 
miles an hour, and the other 3 miles an 
hour, how far apart will they be in 9 hours ? 

66. In how many hours will a man who 
drives 8 miles an hour overtake a footman 
who is 60 miles ahead, and walks at the 
rate of 3 miles an hour? 

67. A man bought a span of horses for 
g100, paid $60 for their keeping, and sold 
them for $200; what did he gain on each 
horse ? 

68. How many turkeys can I buy for 
$43, at the rate of 3 for $5, and have $8 left ? 


EXAMPLES FOR THE SLATE, 


1. In a certain church 28 pews rent at 
$35 each, 19 at $25 each and 37 at $15 
each ; for how much do they all rent? 

2. A railroad 18 miles long cost $452, 
682 for labor, and $177,228 for other ex- 
penses; what was the cost per mile? 

3. One half of the inhabitants of Con- 
stantinople are Turks, 150,000 Greeks, 
30,000 Armenians, and 65,000 Jews; how 
many in all? 

See Teachers’ Fdition, p. 165. 


126 FIRST STEPS AMONG FIGURES. 


4. If a man earns $960 a year, and 
spends yearly $688, in how many years 
will he lay up $4,624. 

5. A man bought a farm for $17,600 ; 
he sold half of it for $9,322, at the rate of 
$79 an acre; how many acres did he buy ? 
How much did he pay an acre? 

6. From the sum of 7574 and 10746, 
take their difference. 

7, A lady having $125, paid $37 fora 
set of furs, and $2 a yard for 23 yards of 
silk ; how much money had she left ? 

8. 133416 emigrants arrived in New 
York in 1867, which was 9,731 more than 
arrived in 1866 ; how many arrived in 1866? 

Q James and George started together, 
and traveled in the same direction. James 
walked 2 miles an hour and George 4 miles 
an hour ; how far apart were they at the end 
of 19 hours? 

10. In six boxes of crayons there are 864 
pieces ; if 864 pieces cost 360 cents, what 
will one box cost ? 

11. There are two numbers, the greater 
of which is 37x96, and their difference is 
18 x 27; what are the numbers? 

12. Aearns $45 a month, and B earns 13 
times as much lacking $490 ; how much does 
B earn in 8 months? 


FIRST STEPS AMONG FIGURES. 127 


13. If a house is worth $1,800 and the 
farm on which it stands five times as much 
lacking $36, and the stock one-third as 
much as the house and farm, what is the 
value of the whole? 

14. A man sold his farm of 245 acres at 
$69 an acre and bought some land at $97 
an acre; how many acres could he buy ? 

15. Mr. Smith was 968 miles from home; 
he traveled toward home 137 miles one day; 
119 the next day, and 98 the third day; 
how far was he from home then ? 

16. From thirty billion ten thousand, 
take seven billion two hundred nine thou- 
sand seventy-five. 

17. A man bought 325 bushels of barley 
for $500; 450 bushels of oats for $250; 
625 bushels of corn for $150 more than he 
paid for the oats ; 300 bushels of beans at 
$2 a bushel, and some wheat for $100 more 
than he paid for the corn ; how much did 
he pay for all? 

18. How many solid feet of earth can be 
removed in 36 days by two carts each 
carrying 9 loads a day, and 34 solid feet at 
a load? 

19. A man having $0,840, gave each of 
his two sons $2,750 and the remainder to 
his daughter; how much did he give his 
daughter ? 


128 FIRST STEPS AMONG FIGURES. 


20. $26,250 is 3 times what A gave for 
his farm, and he gave $370 more for it than 
it was worth; how much was the farm 
worth ? | 

21. I sold a horse for $375 which cost me 
$295 ; how much did I gain? 

22. I sold a cow for $65 and by so doing 
lost $15 ; what did she cost? 

23. A man began business with $3,850, 
and in 7.years he was worth $10,465 ; how 
much did he make each year ? 

24. How many days would 36 horses live 
on an amount of food that would keep 24 
horses 288 days? 

25. A merchant received $248 on Mon- 
day and $396 on Tuesday; what was the 
average receipts per day? 

26. Two men start from the same place 
and travel tn opposite directions, one at the 
rate of 54 miles a day, and the other at the | 
rate of 45 miles a day ; how far apart will 
they be at the end of 6 days? How far 
apart if they travel in the same direction? 

27. A man bought 478 bushels of corn ; 
all but 136 bushels were sunk in a boat; 
how much was saved ? 

28. A merchant bought 46 yards of cloth 
for $93, and sold it at $3 a yard; how much 
did he gain? 3 

See Teachers’ Edition, p. 166. 


FIRST STEPS AMONG FIGURES. 129 


29. Divide the product of 79 and 237 
by their difference. 

30. At $135 each, how many horses 
can be bought for $9,368 ? 

31. How many times can 317 be sub 
tracted from 13,314? 

32. There are 3 bins containing 856 
bushels of wheat ; I contains 376 bushels, 
another contains 297 bushels; how many 
in the third bin? 

33. A farmer sold 13 tons of hay at $16a 
ton, and 24 cords of wood at $5 a cord; he 
divided the money received among four credi- 
tors; how much money did each receive ? 

34. A has 18 barrels of flour of 196 
pounds each ; if a family of 9 persons use 
49 pounds of flour a week, how long will 
the flour last them? 

35. If Mr. Long’s sheep were put into 
6 fields, 96 in a field, there would be 5 
sheep remaining ; how many sheep has he? 

36. A grocer bought 2 cheeses, one 
weighing 68 pounds and the other 75 
pounds, at 14 cents a pound; how many 
cents would he gain by selling them at 17 
cents a pound? 

37. A man killed four hogs, one weigh- 
ing 368 pounds, one 412, one 379 and one 
433; what was their average weight ? 


od 


130 FIRST STEPS AMONG FIGURES. 


38. There were 84 sheep in four pastures; 
there were 30 in the first and 24 in the 
second; if there were an equal number in 
each of the others, how many in each ? 

39. If a man paid $500 for four horses, 
$200 for 5 cows and $175 for 40 sheep, 
how many animals did he buy ? 

4o. If a man earns $685 a year, and 
spends $496 a year, in how many years 
will he save $1,134? 

41. How many pounds of coffee at 27 
cents a pound will pay for three hogsheads 
of sugar, each containing 1080 pounds, at 
I2 cents a pound? 

42. What is the sum of the difference 
and sum of 1768 and 987 ? 

43 Aman deposited in bank at different 
times $397, $459 and $568 ; he drew out 
at one time $275 and at another $368 ; 
how much remained in the bank ? 

44. A man sold 26 cows at $35 each; 
how many horses at $145 each can he buy 
with the money received ? 

45. A dealer shipped 500 bushels of 
beans in 250 bags, 600 bushels of wheat in 
280 bags; he used 136 less bags in which 
to ship 300 bushels of corn than he did for 
the wheat; he put 1200 bushels of oats in 
bags holding 2 bushels each; how many 
bags did he use for all the grain ? 


FIRST STEPS AMONG FIGURES. 13) 


46. The income of a man who “struck 
oil” is $75 a day; how many teachers 
would this employ at $850 a year ? 

47. A farmer having $1397, bought 9 
tons of hay at $16 a ton, a horse for $185, 
155 sheep at $4 each, and spent the rest of 
his money for cows at $32 each ; how many 
cows did he buy? |. 

48. A fisherman caught 2 dozen fishes ; 
he sold one-half of them at 25 cents each ; 
the other half for 26 cents each, except one 
which, weighing 33 pounds, he retailed at 11 
cents per pound; how much did he get for 
his fishes ? 

49. A boy paid 100 cents for 5 quires of 
paper (24 sheets each) and sold it at the 
rate of 2 sheets for 3 cents ; did he gain or 
lose, and how much ? 

50. How many half dimes in 350 cents? 

51. A miller ground 34 bushels of wheat, 
18 of corn, and 22 of oats; how many bags 
holding 2 bushels each, held the grain ? What 
did the grinding cost at 7 cents a bushel ? 

52. 24 boys attended the same school, 
but in three different rooms ; 5 were in one 
room, and 8 in another, and if the number 
of boys in the third room be multiplied by 
12, the product will equal the number of 
blackbirds they saw on their way to school ; 
how many did they see? 


132 FIRST STEPS AMONG FIGURES. 


53. Sound travels at the rate of 1090 
feet in a second; at this rate how long 
would it take the report of a cannon to 
reach the moon, which is 240,000 miles 
away (1 mile is 5280 feet) ? 

54. An estate of $14350 was divided 
between a widow and two children; the 
widow’s share was $5450, the son’s $1280 
less than the widow’s, and the daughter 
had the rest; how much did the daughter 
have? 

55. The product of two numbers is 
36288, and one of them is 756; what is the 
other ? 

56. A man bought 145 acres of land for 
$9,850, and 95 more acres at $45 an acre; he 
sold the whole at $56 an acre; did he gain or 
’ lose, and: how much ? 

57. A farmer bought 47 acres of land 
for $4,416, and 34 acres at $75 an acre; 
what was the average price per acre? 

58. The sum of two numbers is 7568, 
and one of them is 784; what is the other? 

59. How many military companies of 98 
men each, can be formed from 7,463 men? 

60. How many yards of cloth at 24 
cents a yard, will pay for 26 dozen eggs at 
14 cents a dozen, and a jar of butter worth 
284 cents? 


FIRST STEPS AMONG FIGURES. 133 


61. George and Lewis start from the 
same place at the same time, and travel in 
the same direction, George at the rate of 
714 rods an hour, and Lewis at the rate of 
579 rods an hour; how far apart are they 
at the end of g hours? How far apart in 
7 hours, if they had traveled in opposite 
directions ? 

62. If I receive $40 a month and spend 
$32 a*month, in how many years will I 
save $1,152? 

63. Subtract the difference between 79 
and 2300 from their sum. 

64. What is the sum of ten thousand 
ninety, seven thousand nine hundred, eight 
million nine hundred eighteen, five hun- 
dred thousand, seventy thousand seventy- 
five, and eight hundred. 

65. The dividend is 7365092, the divisor is 
6978 ; what is the quotient and remainder ? 

66. The remainder is 658 and the sub- 
trahend 1734; what is the minuend ? 

Pupils make and solve the following ex- 
amples: 

67. Given a multiplicand of 4 figures, a 
multiplier of 3 figures, required the product ? 

68. Given the minuend and remainder, 
find the subtrahend. 

69. Given the subtrahend and the re- 
mainder, find the minuend. 


134 FIRST STEPS AMONG FIGURES. 


70. Given the sum of three numbers and 
two of them, find the third. 

71. Given the difference between two 
numbers and the less number, find the 
greater. 

72. Given the divisor, quotient and re- 
mainder, find the dividend. 

73. Given the product of two numbers. 
and one of them, find the other. 

74. Given the difference between two 
numbers and the greater number, find the 
less number. 

75. Given whole price, number of arti- 
cles, find the price of a different number 
of articles. 

76. Given the cost and selling price, find 
the gain. 

77. Given the selling price and the loss, 
find the cost 

78. Given the cost and the gain, find 
selling price. 

79. Given the selling price and gain, fine 
cost. 

So Given the cost and the loss, find the 
selling price. 


FIRST STEPS AMONG FIGURES. 135 


» 6745 2. 5847 3. 9337 


5678 9576 4598 
9867 4684 3765 
6543 5967 9458 
7698 8439 7695 
4759 4785 4739 
5978 9478 5345 
9047 4567 4869 
8458 8975 3765 
4790 | 4687 4798 
4. 8451 5. 5747 
6759 9835 
7846 4696 
9567 8739 
6976 5684 
7569 7938 
8427 6456 
9508 8845 
3753 7587 


136 


FIRST STEPS AMONG FIGURES. 


7. 4768 8. 5869 
9535 9768 
9849 4352 
7697 4675 
4857 9538 
9564 4678 
5739 5765 
5647 4976 
9788 9738 
5347 4657 
8234 9876 

10. 7465 
es 25 
4795 
8649 
3578 
9756 
6549 
8732 
5685 
7839 


FIRST STEPS AMONG FIGURES. £37 


meaese . 12.99038 |) 13.9768 


9764 5796 7596 
3797 4579 4879 
8436 8947 9087 
9768 + 6896 4739 
4593 . 9758 8645 
8945 4563 3787 
7037 9755 9568 
4895 6879 4837 
7537 5768 8795 
4885 8597 4568 
5938 4856 9756 
9457 9589 | 6897 
14. 6795 15. 5896 
4569 9748 
8769 7635 
9245 5864 
8876 9787 
6789 4538 
3954 7087 
8450 4859 
3789 5321 
8375 4978 
9999 9654 
3478 7987 


6457 9868 


——— 
ee 


138 


16. 


FIRST STEPS AMONG FIGURES. 


17. 9768 
4593 
7846 
5937 + 
9876 
2345 
3527 
8498 
6549 
7856 
7387 
9765 
3849 
9674 
3758 
6847 


eee 


18. 4879. 
9763 
4598 
8756 
4975 
68 14 
7589 
4837 
5692 
7859 
4537 
6895 
4576 
3844 
7989 
4596 


FIRST STEPS AMONG FIGURES. 139 


19. 4532 20. 9476 
7856 3869 
4978 5458 
9456 7507 
3379" 4835 
4856 8769 
7995 7654 
8447 8579 
9568 6432 
4789 4976 
6435 9845 
7896 5637 
4967 7948 
8538 8654 
7689 6739 


9876 4798 


140 FIRST STEPS AMONG FIGURES. 


MULTIPLICATION TABLE. 72’sand review. 


a b € 
8. 910 I1 12. 8 OF IO igo 
9 12.8 10:7 11. 9 125350 eee 


72 108 80 110 84 88 81 120 88 120 56 99 
d vp 


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9 12 810.7, Il Oe 12aGstO eee 


9O 132 96 80 63 I10 99 144 64 64 70 121 
£ h 
3.5212. 8 -O ole a1 Sige 
12. 12).-.0.12 “8.310% 7 ie 
36 00.108 96 72 100 77 1320,R ae 
DIVISION TABLE. 
a b c 
70 I2I 108 96 72 100 132 77 80 110 84 88 
LO. -LIlfpsl anes 20 pete 12 II 10 1 hy or 
RIL: 9123-10. 1427 See 
ad e 
120 108 72 132 90 99 56 88 120 81 96 80 
I2 ‘9. 8 1110 9 8 Ii 103g 
10.12 9 12: QL! 7 6) ee 
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72 48 63 I10 99 144 64 90 36 60 
12 12.9. 10 11512) 65)0nig aim 
64°77 "11 “9% 42-8) toa 
See Teachers’ Edition, p. 156. 


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* 


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